Question

How many subsets of size $4$of the set localid="1649163905451" role="math" $S=\left\{1,2,...,20\right\}$contain at least one of the elements $1,2,3,4,5$?

Short answer

The possible no. of subsets are$3480$.

Step-by-step solution

Step 1. Given information.

The given set is$S=\left\{1,2,...,20\right\}$

The object is to find the number of subsets of size $4$which contains at least one of the elements $1,2,3,4,5$.

Step 2. Find the number of subsets.

No. of ways to select $4$numbers out of $20$isrole="math" localid="1649164747337" $\left(\begin{array}{c}20\\ 4\end{array}\right)=\frac{20!}{4!16!}=4845$

The set of size $4$must contain at least one of the elements $1,2,3,4,5$. So, the remaining numbers in set S will be $15$.

No. of ways to select $4$numbers out of $15$is role="math" localid="1649164852115" $\left(\begin{array}{c}15\\ 4\end{array}\right)=\frac{15!}{4!11!}=1365$

Therefore, the possible no. of subsets are$=4845-1365=3480$