Question

Let $\bigcup_{i=1}^{50} X_i = \bigcup_{i=1}^{n} Y_i = T$, where each $X_i$ contains 10 elements and each $Y_i$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X_i$'s and exactly 6 of sets $Y_i$'s then $n$ is equal to :

• A. 30
• B. 25
• C. 150
• D. 50

Answer 1

Answer: (A)

30

Answer 2

The total number of elements counted across all sets $X_i$ is $50 \times 10 = 500$. Since each element of $T$ belongs to exactly 20 of these sets, the total number of elements in $T$ is $|T| = \frac{500}{20} = 25$.

The total number of elements counted across all sets $Y_j$ is $n \times 5 = 5n$. Since each element of $T$ belongs to exactly 6 of these sets, the total count can also be expressed as $6 \times |T|$.

Equating the two expressions for the total count related to $T$: $5n = 6 \times |T|$ Substitute $|T| = 25$: $5n = 6 \times 25$ $5n = 150$ $n = \frac{150}{5}$ $n = 30$