Solution: 2014 Winter Midterm - 9

There are 13 letters in the word “POOPERSCOOPER”.

We can break down how many letters of each there are:

• 3 P's
• 4 O's
• 2 E's
• 2 R's
• 1 C
• 1 S

• We choose 1 C from the 13 spots = $\binom{13}{1}$
• We choose 4 O's from the remaining 12 spots = $\binom{12}{4}$
• We choose 2 E's from the remaining 8 spots = $\binom{8}{2}$
• We choose 1 S from the remaining 6 spots = $\binom{6}{1}$
• We choose 2 R's from the remaining 5 spots = $\binom{5}{2}$
• We choose 3 P's from the remaining 3 spots = $\binom{3}{3}$

Thus, the total number of strings is

$\binom{13}{1} \cdot \binom{12}{4} \cdot \binom{8}{2} \cdot \binom{6}{1} \cdot \binom{5}{2} \cdot \binom{3}{3}$