A. Find the number of distinguishable permutations of the given letters. (Leave answers in factorial notation.) $$ \begin{array}{ll} ext { 1) AAAB B C } \ ext { 2) } \ ext { 3ABBCC } \ ext { 3) AAABBBCCCD } \ ext { 4) } 717676711\end{array} $$

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A. Find the number of distinguishable permutations of the given letters. (Leave answers in factorial notation.) $$ \begin{array}{ll}	ext { 1) AAAB B C } \ 	ext { 2) } \ 	ext { 3ABBCC } \ 	ext { 3) AAABBBCCCD } \ 	ext { 4) } 717676711\end{array} $$

UpStudy AI · Accepted Answer

1) The word is $$ \text{AAABBC} $$. There are 6 letters in all with frequencies  
  $$ A: 3,\quad B: 2,\quad C: 1 $$.  
The number of distinguishable arrangements is  
$$
\frac{6!}{3!2!1!} = \frac{6!}{3!2!}.
$$

2) The word is $$ \text{AABBCC} $$. There are 6 letters with counts  
  $$ A: 2,\quad B: 2,\quad C: 2 $$.  
The number of distinguishable arrangements is  
$$
\frac{6!}{2!2!2!}.
$$

3) The word is $$ \text{AAABBBCCCD} $$. There are 10 letters with  
  $$ A: 3,\quad B: 3,\quad C: 3,\quad D: 1 $$.  
The number of distinguishable arrangements is  
$$
\frac{10!}{3!3!3!1!} = \frac{10!}{3!3!3!}.
$$

4) The string is $$ 717676711 $$. There are 9 digits with frequency counts  
  $$ 7: 4,\quad 1: 3,\quad 6: 2 $$.  
The number of distinguishable arrangements is  
$$
\frac{9!}{4!3!2!}.
$$
1) $$ \frac{6!}{3!2!} $$ 2) $$ \frac{6!}{2!2!2!} $$ 3) $$ \frac{10!}{3!3!3!} $$ 4) $$ \frac{9!}{4!3!2!} $$

A. Find the number of distinguishable permutations of the given letters. (Leave answers in
factorial notation.)

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