Question

Let X be the set of first 100 natural numbers. Sets S1 and S2 are subsets of X such that each of them has more than zero elements and no common element. If the maximum number of elements in S1 = x1, in S2 = y1 and minimum no. of elements in S1 = x2, in S2 = y2 then:
Suppose the union of sets S1 and S2 is X and S1, S2 have the same number of elements. If elements from S1 and S2 are chosen randomly, it was found that by exchanging two elements of S1 with two elements of S2, every element of S1 was greater than that of S2. Find the no. of sets S1 that can be formed.

• A. ⁵⁰C₂
• B. ⁵⁰C₄₉
• C. 100
• D. ⁵⁰C₄₈ * ⁵⁰C₂

Answer 1

The correct option is D

⁵⁰C₄₈ * ⁵⁰C₂

In set S1, 48 elements can be selected randomly in ${}^{50}{C}_{48}$ ways (from 51 to 100) and the other two elements in S2 can be selected in ${}^{50}{C}_2$ ways (from 1 to 50). So, set S1 can be formed in ${}^{50}{C}_{48}{\times }^{50}{C}_2$ ways. Hence, option (d).