Let ∪ ᵢ₌₁⁵⁰ Xᵢ = ∪ ᵢ₌₁ⁿ Yᵢ = T where each Xᵢ contains 10 elements and each Yᵢ contains 5 elements. If each element of the set T is an element of exactly 20 of sets Xᵢ’s and exactly 6 of sets Yᵢ’s, then n is equal to :

Solution

Maths sir · 3 steps

Count the memberships two ways: once set by set, once element by element, and |T| drops out.

1. Count X memberships. Add the sizes of the 50 sets Xᵢ: 50 × 10 = 500 memberships. Each element of T sits in exactly 20 of the Xᵢ, so the same total is 20|T|.

   20|T| = 500 ⇒ |T| = 25

2. Count Y memberships. The n sets Yᵢ have 5n memberships in total. Each of the 25 elements lies in exactly 6 of them, so the total is also 6 × 25 = 150.
3. So. 5n = 150 gives n = 30, which is option (a).

The idea to keep. When every element lies in a fixed number of sets, use double counting: (sum of set sizes) = (number of elements) × (sets per element). It turns overlapping unions into one equation.

Answer from the NTA final answer key. Solution by PrepWiser.