Question

Let $1\le m<n\le p$. The number of subsets of the set $A=\left\{1,2,3,........p\right\}$ having $m,n$ as the least and the greatest elements respectively, is

• A. $2^{n-m-1}-1$
• B. $2^{n-m-1}$
• C. $2^{n-m}$
• D. $2^{p-n+m-1}$

Answer 1

The correct option is B

$2^{n-m-1}$

Explanation for correct option:

Finding the number of subsets:

Given, $1\le m<n\le p$ and set $A=\left\{1,2,3,....p\right\}$

Now the number of integers between $m$ and $n$ is equal to $n-m-1$

Now each of the $n-m-1$elements has two choices that is, they will be selected or will not be selected.

Therefore the number of subsets of set $A$having $m$ and $n$ as least and greatest elements respectively is equal to, $2^{n-m-1}$

Hence, the correct answer is option (B).