Solution: 2019 Winter Midterm - 4

Author: Michiel Smid

Question

Let $n \geq 8$ be an even integer and let $k$ be an integer with $2 \leq k \leq n/2$. Consider $k$-element subsets of the set $S = \{1,2,\dots,n\}$. How many such subsets contain at least two even numbers?

(a)

${n \choose k} - {n/2 \choose k} - \frac{n}{2} \cdot {n/2 \choose k}$

(b)

${n \choose k} - {n/2 \choose k - 1} - \frac{n}{2} \cdot {n/2 \choose k}$

(c)

${n \choose k} - {n/2 \choose k} - \frac{n}{2} \cdot {n/2 \choose k - 1}$

(d)

${n \choose k} - {n/2 \choose k - 1} - \frac{n}{2} \cdot {n/2 \choose k - 1}$

Solution

$ \text{All Possibilities} - \text{0 even numbers} - \text{1 even number} $

$ \binom{n}{k} - \binom{n/2}{k} - \binom{n/2}{k-1} $