Solution: 2015 Fall Final - 2

Author: Michiel Smid

Question

Consider a set $S$ consisting of 20 integers. The integer 0 is an element of $S$, 9 elements in $S$ are strictly positive, and the remaining 10 elements are strictly negative. What is the number of 7-element subsets of $S$ having the property that the product of the 7 elements in the subset is equal to 0?

(a)

${19 \choose 6}$

(b)

${20 \choose 6}$

(c)

${19 \choose 7}$

(d)

${20}\choose{7}$

Solution

Since the product of the 7 elements in the subset is 0, the subset must contain 0 once: 1

From here, it doesn’t matter which numbers we choose since multiplying values by 0 will result in 0: $ \binom{19}{6} $