Solution: 2018 Fall Midterm - 7

Author: Michiel Smid

Question

Consider 5-element subsets $\{x_1,x_2,x_3,x_4,x_5\}$ of the set $\{1,2,3,\dots,17\}$, where $x_1 < x_2 < x_3 < x_4 < x_5$.
How many such subsets have the property that $x_3 = 7$?

(a)

${7 choose 2} cdot {9 choose 2}$

(b)

${6 choose 2} cdot {9 choose 2}$

(c)

${6 choose 2} cdot {10 choose 2}$

(d)

${7 choose 2} cdot {10 choose 2}$

Solution

If $ x_3 = 7 $, then we need to choose 2 elements from the set $ {1, 2, 3, 4, 5, 6} $ for $ x_1 $ and $ x_2 $

There are $ \binom{6}{2} $ ways to choose 2 elements from the set $ {1, 2, 3, 4, 5, 6} $

We need to choose 2 elements from the set $ {8, 9, 10, 11, 12, 13, 14, 15, 16, 17} $ for $ x_4 $ and $ x_5 $

There are $ \binom{10}{2} $ ways to choose 2 elements from the set $ {8, 9, 10, 11, 12, 13, 14, 15, 16, 17} $

Thus, there are $ \binom{6}{2} \cdot \binom{10}{2} $ such subsets.