Length-$8$ rearrangements of AABBCCDD with pairs AA, BB, CC adjacent The Next CEO of Stack OverflowNumber of words with length 8 with certain restrictionsPermutation of 4 letters in a 11-letter word : how many different words can we make?Differentiating between rearrangements and permutations.Number of possible combinations of the Enigma machine plugboardLetter combinatorics and probabilitiesNumber of words with length 8 with certain restrictions$42$ letters on a typewriterNumber of SequenceFind the number of eight-letter words that use letters from the set A, B, C and contain exactly three As…Words of length $10$ are formed using the letters A, B, C, D, E, F, G, H, I, JEnvelope Question: Five letters addressed to individuals 1-5 are randomly placed in five addressed envelopes, one letter in each envelope.

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Length-$8$ rearrangements of AABBCCDD with pairs AA, BB, CC adjacent



The Next CEO of Stack OverflowNumber of words with length 8 with certain restrictionsPermutation of 4 letters in a 11-letter word : how many different words can we make?Differentiating between rearrangements and permutations.Number of possible combinations of the Enigma machine plugboardLetter combinatorics and probabilitiesNumber of words with length 8 with certain restrictions$42$ letters on a typewriterNumber of SequenceFind the number of eight-letter words that use letters from the set A, B, C and contain exactly three As…Words of length $10$ are formed using the letters A, B, C, D, E, F, G, H, I, JEnvelope Question: Five letters addressed to individuals 1-5 are randomly placed in five addressed envelopes, one letter in each envelope.










3












$begingroup$


I am doing bigger task from combinatorics and stuck at this sub-problem:




Find number of words which every letter A, B, C, D occurs exactly 2 times and exactly 3 pairs of same letters occurs on neighbouring positions




Ok, my approach was just writing some cases and trying to find some rules for that. But I find interesting way to do this without big explanation and I am not sure why it works



Solution



$$ binom43underbracebinom51_textAAunderbracebinom41_textBBunderbracebinom31_textCC $$



My current understanding



$binom43 $ - we choose 3 types of letters from 4 available
$underbracebinom51_textAA$ we choose... position for (for example) $A$, probably i'th place and we put to $(i,i+1)$ first letter. But why it is
$$ underbracebinom51_textAA$$
And not $$ underbracebinom71_textAA$$
Can somebody explain me this approach? It seems interesting and really smart.










share|cite|improve this question











$endgroup$











  • $begingroup$
    We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
    $endgroup$
    – N. F. Taussig
    2 days ago











  • $begingroup$
    @N.F.Taussig That's an answer. Why are you posting it in the comment section?
    $endgroup$
    – Arthur
    2 days ago











  • $begingroup$
    @Arthur I am questioning the posted solution.
    $endgroup$
    – N. F. Taussig
    2 days ago






  • 1




    $begingroup$
    This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
    $endgroup$
    – VirtualUser
    2 days ago
















3












$begingroup$


I am doing bigger task from combinatorics and stuck at this sub-problem:




Find number of words which every letter A, B, C, D occurs exactly 2 times and exactly 3 pairs of same letters occurs on neighbouring positions




Ok, my approach was just writing some cases and trying to find some rules for that. But I find interesting way to do this without big explanation and I am not sure why it works



Solution



$$ binom43underbracebinom51_textAAunderbracebinom41_textBBunderbracebinom31_textCC $$



My current understanding



$binom43 $ - we choose 3 types of letters from 4 available
$underbracebinom51_textAA$ we choose... position for (for example) $A$, probably i'th place and we put to $(i,i+1)$ first letter. But why it is
$$ underbracebinom51_textAA$$
And not $$ underbracebinom71_textAA$$
Can somebody explain me this approach? It seems interesting and really smart.










share|cite|improve this question











$endgroup$











  • $begingroup$
    We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
    $endgroup$
    – N. F. Taussig
    2 days ago











  • $begingroup$
    @N.F.Taussig That's an answer. Why are you posting it in the comment section?
    $endgroup$
    – Arthur
    2 days ago











  • $begingroup$
    @Arthur I am questioning the posted solution.
    $endgroup$
    – N. F. Taussig
    2 days ago






  • 1




    $begingroup$
    This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
    $endgroup$
    – VirtualUser
    2 days ago














3












3








3





$begingroup$


I am doing bigger task from combinatorics and stuck at this sub-problem:




Find number of words which every letter A, B, C, D occurs exactly 2 times and exactly 3 pairs of same letters occurs on neighbouring positions




Ok, my approach was just writing some cases and trying to find some rules for that. But I find interesting way to do this without big explanation and I am not sure why it works



Solution



$$ binom43underbracebinom51_textAAunderbracebinom41_textBBunderbracebinom31_textCC $$



My current understanding



$binom43 $ - we choose 3 types of letters from 4 available
$underbracebinom51_textAA$ we choose... position for (for example) $A$, probably i'th place and we put to $(i,i+1)$ first letter. But why it is
$$ underbracebinom51_textAA$$
And not $$ underbracebinom71_textAA$$
Can somebody explain me this approach? It seems interesting and really smart.










share|cite|improve this question











$endgroup$




I am doing bigger task from combinatorics and stuck at this sub-problem:




Find number of words which every letter A, B, C, D occurs exactly 2 times and exactly 3 pairs of same letters occurs on neighbouring positions




Ok, my approach was just writing some cases and trying to find some rules for that. But I find interesting way to do this without big explanation and I am not sure why it works



Solution



$$ binom43underbracebinom51_textAAunderbracebinom41_textBBunderbracebinom31_textCC $$



My current understanding



$binom43 $ - we choose 3 types of letters from 4 available
$underbracebinom51_textAA$ we choose... position for (for example) $A$, probably i'th place and we put to $(i,i+1)$ first letter. But why it is
$$ underbracebinom51_textAA$$
And not $$ underbracebinom71_textAA$$
Can somebody explain me this approach? It seems interesting and really smart.







combinatorics discrete-mathematics combinations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 days ago









user21820

39.9k544159




39.9k544159










asked 2 days ago









VirtualUserVirtualUser

1,183217




1,183217











  • $begingroup$
    We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
    $endgroup$
    – N. F. Taussig
    2 days ago











  • $begingroup$
    @N.F.Taussig That's an answer. Why are you posting it in the comment section?
    $endgroup$
    – Arthur
    2 days ago











  • $begingroup$
    @Arthur I am questioning the posted solution.
    $endgroup$
    – N. F. Taussig
    2 days ago






  • 1




    $begingroup$
    This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
    $endgroup$
    – VirtualUser
    2 days ago

















  • $begingroup$
    We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
    $endgroup$
    – N. F. Taussig
    2 days ago











  • $begingroup$
    @N.F.Taussig That's an answer. Why are you posting it in the comment section?
    $endgroup$
    – Arthur
    2 days ago











  • $begingroup$
    @Arthur I am questioning the posted solution.
    $endgroup$
    – N. F. Taussig
    2 days ago






  • 1




    $begingroup$
    This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
    $endgroup$
    – VirtualUser
    2 days ago
















$begingroup$
We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
$endgroup$
– N. F. Taussig
2 days ago





$begingroup$
We have five objects to arrange. In your example, they are AA, BB, CC, D, D. However, the posted solution does not account for the requirement that exactly $3$ pairs of identical letters are in neighboring positions.
$endgroup$
– N. F. Taussig
2 days ago













$begingroup$
@N.F.Taussig That's an answer. Why are you posting it in the comment section?
$endgroup$
– Arthur
2 days ago





$begingroup$
@N.F.Taussig That's an answer. Why are you posting it in the comment section?
$endgroup$
– Arthur
2 days ago













$begingroup$
@Arthur I am questioning the posted solution.
$endgroup$
– N. F. Taussig
2 days ago




$begingroup$
@Arthur I am questioning the posted solution.
$endgroup$
– N. F. Taussig
2 days ago




1




1




$begingroup$
This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
$endgroup$
– VirtualUser
2 days ago





$begingroup$
This solution follows from math.stackexchange.com/a/1828004/617243 I have solved singles, and pairs on my own, but stucked on triples (too many cases to just write them all) and seen linked solution.
$endgroup$
– VirtualUser
2 days ago











2 Answers
2






active

oldest

votes


















3












$begingroup$

The count you gave from the linked question is for at least three pairs of identical letters in adjacent positions in an Inclusion-Exclusion Principle argument.



There are $binom43$ ways of choosing which three pairs of identical letters are in adjacent positions. Suppose, as in your example, they are AA, BB, and CC. Then we have five objects to arrange. They are AA, BB, CC, D, and D. The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. The Ds must be placed in the remaining two positions. That gives the count
$$binom43binom51binom41binom31$$
you found in the linked problem. Notice, however, that this includes arrangements such as
$$AADDCCBB$$
in which there are four pairs of adjacent identical letters. There are $4!$ such arrangements for each of the $binom43$ ways we could designate three of the four pairs as the three identical pairs. Hence, the number of arrangements with exactly three pairs of adjacent identical letters is
$$binom43binom51binom41binom31 - binom434!$$



Another way to see this is to choose three of the four letters to be the adjacent identical pairs, which can be done in $binom43$ ways. The chosen pairs can be arranged in $3!$ ways. This creates four spaces, two between successive pairs and two at the ends of the row. For instance, suppose the chosen pairs are AA, BB, and CC. Then
$$square AA square BB square CC square$$
is one possible arrangement. To ensure that the remaining two identical letters are separated, we must choose two of the four spaces in which to place the remaining letter. Thus, the number of arrangements of A, A, B, B, C, C, D, D with exactly three pairs of adjacent identical letters is
$$binom433!binom42$$






share|cite|improve this answer









$endgroup$












  • $begingroup$
    I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
    $endgroup$
    – VirtualUser
    2 days ago










  • $begingroup$
    As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
    $endgroup$
    – N. F. Taussig
    2 days ago










  • $begingroup$
    Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
    $endgroup$
    – VirtualUser
    2 days ago











  • $begingroup$
    That is correct.
    $endgroup$
    – N. F. Taussig
    2 days ago










  • $begingroup$
    Okay, thanks for explanation
    $endgroup$
    – VirtualUser
    2 days ago


















2












$begingroup$

There are $4$ choices for the nonconsecutive letter.



Let's examine the case where $DD$ does not occur.



There are $3!$ orders of the sort: $$cdot AAcdot BBcdot CCcdot$$ where $2$ of the dots must be filled in with a letter $D$.



So there are $binom42$ possibilities for placing the letters $D$.



So we arrive at:$$4times3!timesbinom42=4times6times6=144$$solutions.






share|cite|improve this answer









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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    The count you gave from the linked question is for at least three pairs of identical letters in adjacent positions in an Inclusion-Exclusion Principle argument.



    There are $binom43$ ways of choosing which three pairs of identical letters are in adjacent positions. Suppose, as in your example, they are AA, BB, and CC. Then we have five objects to arrange. They are AA, BB, CC, D, and D. The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. The Ds must be placed in the remaining two positions. That gives the count
    $$binom43binom51binom41binom31$$
    you found in the linked problem. Notice, however, that this includes arrangements such as
    $$AADDCCBB$$
    in which there are four pairs of adjacent identical letters. There are $4!$ such arrangements for each of the $binom43$ ways we could designate three of the four pairs as the three identical pairs. Hence, the number of arrangements with exactly three pairs of adjacent identical letters is
    $$binom43binom51binom41binom31 - binom434!$$



    Another way to see this is to choose three of the four letters to be the adjacent identical pairs, which can be done in $binom43$ ways. The chosen pairs can be arranged in $3!$ ways. This creates four spaces, two between successive pairs and two at the ends of the row. For instance, suppose the chosen pairs are AA, BB, and CC. Then
    $$square AA square BB square CC square$$
    is one possible arrangement. To ensure that the remaining two identical letters are separated, we must choose two of the four spaces in which to place the remaining letter. Thus, the number of arrangements of A, A, B, B, C, C, D, D with exactly three pairs of adjacent identical letters is
    $$binom433!binom42$$






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
      $endgroup$
      – VirtualUser
      2 days ago










    • $begingroup$
      As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
      $endgroup$
      – VirtualUser
      2 days ago











    • $begingroup$
      That is correct.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Okay, thanks for explanation
      $endgroup$
      – VirtualUser
      2 days ago















    3












    $begingroup$

    The count you gave from the linked question is for at least three pairs of identical letters in adjacent positions in an Inclusion-Exclusion Principle argument.



    There are $binom43$ ways of choosing which three pairs of identical letters are in adjacent positions. Suppose, as in your example, they are AA, BB, and CC. Then we have five objects to arrange. They are AA, BB, CC, D, and D. The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. The Ds must be placed in the remaining two positions. That gives the count
    $$binom43binom51binom41binom31$$
    you found in the linked problem. Notice, however, that this includes arrangements such as
    $$AADDCCBB$$
    in which there are four pairs of adjacent identical letters. There are $4!$ such arrangements for each of the $binom43$ ways we could designate three of the four pairs as the three identical pairs. Hence, the number of arrangements with exactly three pairs of adjacent identical letters is
    $$binom43binom51binom41binom31 - binom434!$$



    Another way to see this is to choose three of the four letters to be the adjacent identical pairs, which can be done in $binom43$ ways. The chosen pairs can be arranged in $3!$ ways. This creates four spaces, two between successive pairs and two at the ends of the row. For instance, suppose the chosen pairs are AA, BB, and CC. Then
    $$square AA square BB square CC square$$
    is one possible arrangement. To ensure that the remaining two identical letters are separated, we must choose two of the four spaces in which to place the remaining letter. Thus, the number of arrangements of A, A, B, B, C, C, D, D with exactly three pairs of adjacent identical letters is
    $$binom433!binom42$$






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
      $endgroup$
      – VirtualUser
      2 days ago










    • $begingroup$
      As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
      $endgroup$
      – VirtualUser
      2 days ago











    • $begingroup$
      That is correct.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Okay, thanks for explanation
      $endgroup$
      – VirtualUser
      2 days ago













    3












    3








    3





    $begingroup$

    The count you gave from the linked question is for at least three pairs of identical letters in adjacent positions in an Inclusion-Exclusion Principle argument.



    There are $binom43$ ways of choosing which three pairs of identical letters are in adjacent positions. Suppose, as in your example, they are AA, BB, and CC. Then we have five objects to arrange. They are AA, BB, CC, D, and D. The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. The Ds must be placed in the remaining two positions. That gives the count
    $$binom43binom51binom41binom31$$
    you found in the linked problem. Notice, however, that this includes arrangements such as
    $$AADDCCBB$$
    in which there are four pairs of adjacent identical letters. There are $4!$ such arrangements for each of the $binom43$ ways we could designate three of the four pairs as the three identical pairs. Hence, the number of arrangements with exactly three pairs of adjacent identical letters is
    $$binom43binom51binom41binom31 - binom434!$$



    Another way to see this is to choose three of the four letters to be the adjacent identical pairs, which can be done in $binom43$ ways. The chosen pairs can be arranged in $3!$ ways. This creates four spaces, two between successive pairs and two at the ends of the row. For instance, suppose the chosen pairs are AA, BB, and CC. Then
    $$square AA square BB square CC square$$
    is one possible arrangement. To ensure that the remaining two identical letters are separated, we must choose two of the four spaces in which to place the remaining letter. Thus, the number of arrangements of A, A, B, B, C, C, D, D with exactly three pairs of adjacent identical letters is
    $$binom433!binom42$$






    share|cite|improve this answer









    $endgroup$



    The count you gave from the linked question is for at least three pairs of identical letters in adjacent positions in an Inclusion-Exclusion Principle argument.



    There are $binom43$ ways of choosing which three pairs of identical letters are in adjacent positions. Suppose, as in your example, they are AA, BB, and CC. Then we have five objects to arrange. They are AA, BB, CC, D, and D. The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. The Ds must be placed in the remaining two positions. That gives the count
    $$binom43binom51binom41binom31$$
    you found in the linked problem. Notice, however, that this includes arrangements such as
    $$AADDCCBB$$
    in which there are four pairs of adjacent identical letters. There are $4!$ such arrangements for each of the $binom43$ ways we could designate three of the four pairs as the three identical pairs. Hence, the number of arrangements with exactly three pairs of adjacent identical letters is
    $$binom43binom51binom41binom31 - binom434!$$



    Another way to see this is to choose three of the four letters to be the adjacent identical pairs, which can be done in $binom43$ ways. The chosen pairs can be arranged in $3!$ ways. This creates four spaces, two between successive pairs and two at the ends of the row. For instance, suppose the chosen pairs are AA, BB, and CC. Then
    $$square AA square BB square CC square$$
    is one possible arrangement. To ensure that the remaining two identical letters are separated, we must choose two of the four spaces in which to place the remaining letter. Thus, the number of arrangements of A, A, B, B, C, C, D, D with exactly three pairs of adjacent identical letters is
    $$binom433!binom42$$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 days ago









    N. F. TaussigN. F. Taussig

    45k103358




    45k103358











    • $begingroup$
      I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
      $endgroup$
      – VirtualUser
      2 days ago










    • $begingroup$
      As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
      $endgroup$
      – VirtualUser
      2 days ago











    • $begingroup$
      That is correct.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Okay, thanks for explanation
      $endgroup$
      – VirtualUser
      2 days ago
















    • $begingroup$
      I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
      $endgroup$
      – VirtualUser
      2 days ago










    • $begingroup$
      As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
      $endgroup$
      – VirtualUser
      2 days ago











    • $begingroup$
      That is correct.
      $endgroup$
      – N. F. Taussig
      2 days ago










    • $begingroup$
      Okay, thanks for explanation
      $endgroup$
      – VirtualUser
      2 days ago















    $begingroup$
    I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
    $endgroup$
    – VirtualUser
    2 days ago




    $begingroup$
    I understand your second way but still can' realize why The position of AA can be selected in five ways, the position of BB can be selected in four ways, and the position of CC can be selected in three ways. I have $AA$ to put and 8 slots. I choose $i$ on $binom71 $ ways and put $AA$ to $(i,i+1)$
    $endgroup$
    – VirtualUser
    2 days ago












    $begingroup$
    As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
    $endgroup$
    – N. F. Taussig
    2 days ago




    $begingroup$
    As I said, we have five objects to arrange: AA, BB, CC, D, D. Thus, we have five positions to fill. If we place AA first, then we have five ways we can place it.
    $endgroup$
    – N. F. Taussig
    2 days ago












    $begingroup$
    Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
    $endgroup$
    – VirtualUser
    2 days ago





    $begingroup$
    Ok, so you treat these object as single letter? For example $AA$ is just one block, also $BB$ and others. That's why we think about $8$-length word as about $textsize of block AA+ textsize of block BB + textsize of block CC + 2* textsize of block D$ length word?
    $endgroup$
    – VirtualUser
    2 days ago













    $begingroup$
    That is correct.
    $endgroup$
    – N. F. Taussig
    2 days ago




    $begingroup$
    That is correct.
    $endgroup$
    – N. F. Taussig
    2 days ago












    $begingroup$
    Okay, thanks for explanation
    $endgroup$
    – VirtualUser
    2 days ago




    $begingroup$
    Okay, thanks for explanation
    $endgroup$
    – VirtualUser
    2 days ago











    2












    $begingroup$

    There are $4$ choices for the nonconsecutive letter.



    Let's examine the case where $DD$ does not occur.



    There are $3!$ orders of the sort: $$cdot AAcdot BBcdot CCcdot$$ where $2$ of the dots must be filled in with a letter $D$.



    So there are $binom42$ possibilities for placing the letters $D$.



    So we arrive at:$$4times3!timesbinom42=4times6times6=144$$solutions.






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      There are $4$ choices for the nonconsecutive letter.



      Let's examine the case where $DD$ does not occur.



      There are $3!$ orders of the sort: $$cdot AAcdot BBcdot CCcdot$$ where $2$ of the dots must be filled in with a letter $D$.



      So there are $binom42$ possibilities for placing the letters $D$.



      So we arrive at:$$4times3!timesbinom42=4times6times6=144$$solutions.






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        There are $4$ choices for the nonconsecutive letter.



        Let's examine the case where $DD$ does not occur.



        There are $3!$ orders of the sort: $$cdot AAcdot BBcdot CCcdot$$ where $2$ of the dots must be filled in with a letter $D$.



        So there are $binom42$ possibilities for placing the letters $D$.



        So we arrive at:$$4times3!timesbinom42=4times6times6=144$$solutions.






        share|cite|improve this answer









        $endgroup$



        There are $4$ choices for the nonconsecutive letter.



        Let's examine the case where $DD$ does not occur.



        There are $3!$ orders of the sort: $$cdot AAcdot BBcdot CCcdot$$ where $2$ of the dots must be filled in with a letter $D$.



        So there are $binom42$ possibilities for placing the letters $D$.



        So we arrive at:$$4times3!timesbinom42=4times6times6=144$$solutions.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 days ago









        drhabdrhab

        104k545136




        104k545136



























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