Question

Let \[1 \le m < n \le p\]. What is the number of subsets of the set \[A = \left\{ {1,2,3, \cdots ,p} \right\}\] having m, n as the least and greatest elements respectively?
A. \[{2^{n - m - 1}} - 1\]
B. \[{2^{n - m - 1}}\]
C. \[{2^{n - m}}\]
D. \[{2^{p - n + m - 1}}\]

Answer 1

Hint: First we find the number of elements between m and n. Then we will apply subset formula to calculate the subsets of the sets.

Formula Used: If a set has n elements, then the number subsets of the set is \[{2^n}\].

Complete step by step solution: Given that, \[A = \left\{ {1,2,3, \cdots ,p} \right\}\] thus the elements of the set A are a collection of positive integers.
Also given that, \[1 \le m < n \le p\] and m, n are the least and greatest elements of the given set.
Therefore m and n are integers.
The number of positive integers between m and n is \[n - m - 1\].
There are two choices that are \[n - m - 1\] are the elements of subsets or they are not elements of subsets and m and n are must be an elements of the subsets.
The number of subsets of the set having m, n as the least and greatest elements respectively is
\[{2^{n - m - 1}}\].

Option ‘B’ is correct

Note: Students often do a mistake to calculate the number of elements between m and n. They calculate that the number of elements between m and n is n – m which is incorrect. The correct answer is the number of elements between m and n is n – m – 1.