Q. 1.16 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

The possible no. of subsets are $3480$.

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 $20$ is  $\left(\begin{array}{c}20\\ 4\end{array}\right)=\frac{20!}{4!16!}=4845$

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 $\left(\begin{array}{c}15\\ 4\end{array}\right)=\frac{15!}{4!11!}=1365$

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