Question

Let $1\le m<n\le p$. The number of subsets of the set $A=1,2,3,...,p$ 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. none of these

Answer 1

Answer: (B)

$2^{n-m-1}$

So, here $n$ should be the greatest integer and $m$ should be the smallest integer.
So, the subsets should only have the elements $m,n$ and the integers between them.

Number of integers between $m$ and $n$ are $=n-m-1$

Now, each of the $(n-m-1)$ elements have $2$ choices i.e. they will be selected or will not be selected

$\therefore$ number of subsets $=2^{n-m-1}$