Question: 2019 Winter Final - 7

Author: Michiel Smid

A string that is obtained by rearranging the letters of the word

POOPERSCOOPER

is called cool, if each occurrence of E has the letter R to its left or right, and each occurrence of R has the letter E to its left or right. Thus, both

POOPERSCOOPER

and

OPRECSOOOERPP

are cool, whereas

EPOOPRSCOOPER

is not cool. What is the number of cool strings?

(a)

${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 3$

(b)

${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 4$

(c)

${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1}$

(d)

${11 \choose 3} \cdot {8 \choose 4} \cdot {4 \choose 2} \cdot {2 \choose 1} \cdot {1 \choose 1} \cdot 2$