Solution: 2017 Winter Midterm - 10
Author: Michiel SmidQuestion
How many strings can be obtained by rearranging the letters of the word
POOPERSCOOPER
(a)
$13!$
(b)
${13 \choose 4}{9 \choose 3}{6 \choose 2}{4 \choose 2}{2 \choose 1}$
(c)
${13 \choose 4}{13 \choose 3}{13 \choose 2}{13 \choose 2}{13 \choose 1}$
(d)
${13 \choose 4}{9 \choose 3}{6 \choose 2}{4 \choose 2}$
Solution
Let’s count the number of times each letter appears in the word:
- 3 P's
- 4 O's
- 2 E's
- 2 R's
- 1 C
- 1 S
We choose 4 of the 13 positions to place the O’s: $ \binom{13}{4} $: $ \binom{13}{4} $
We choose 3 of the remaining 9 positions to place the P’s: $ \binom{9}{3} $: $ \binom{9}{3} $
We choose 2 of the remaining 6 positions to place the E’s: $ \binom{6}{2} $: $ \binom{6}{2} $
We choose 2 of the remaining 4 positions to place the R’s: $ \binom{4}{2} $: $ \binom{4}{2} $
We choose 1 of the remaining 2 positions to place the C’s: $ \binom{2}{1} $: $ \binom{2}{1} $
We choose the last position to place the single $S$: $ 1 $
Thus, there are $ \binom{13}{4} \cdot \binom{9}{3} \cdot \binom{6}{2} \cdot \binom{4}{2} \cdot \binom{2}{1} \cdot 1 $ strings