Solution: 2017 Winter Midterm - 10

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