Let \mathop \cup \limits_i = 1^50 X_i = \mathop \cup \limits_i = 1^n Y_i = 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 :

Sets and Relations PYQ — JEE Main 2020 · 4 Sep · Shift 2 · Q37 with Solution

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Correct answer: A

Let the total number of elements in the set $T$ be $|T|=m$ .
Count the total number of incidences $(\text{element}, X_i)$ in two ways.
- There are $50$ sets $X_i$ , each containing $10$ elements.
- So total incidences with the $X_i$ 's are $50 \times 10 = 500.$
Also, each element of $T$ belongs to exactly $20$ of the sets $X_i$ . Hence, $20m = 500.$ Therefore, $m = \frac{500}{20} = 25.$
Now count the total number of incidences $(\text{element}, Y_i)$ in two ways.
- There are $n$ sets $Y_i$ , each containing $5$ elements.
- So total incidences with the $Y_i$ 's are $5n.$
Also, each element of $T$ belongs to exactly $6$ of the sets $Y_i$ . Since $|T|=25$ , total incidences are also $25 \times 6 = 150.$

Hence, $5n = 150.$
Solve for $n$ : $n = \frac{150}{5} = 30.$
Checking options:
- A: $30$ ✔
- B: $50$ ✘
- C: $15$ ✘
- D: $45$ ✘

Therefore, the correct answer is A.