© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Given Information
We have 50 sets $X_i$ (for $i = 1, 2, \ldots, 50$), each containing 10 elements. We have $n$ sets $Y_j$ (for $j = 1, 2, \ldots, n$), each containing 5 elements. The union of all $X_i$'s and the union of all $Y_j$'s is the same set $T$.
Each element in $T$ belongs to exactly 20 of the $X_i$ sets and exactly 6 of the $Y_j$ sets. We need to find the value of $n$.
Step 2: Compute the Total “Membership Count” for $X_i$ Sets
Since there are 50 sets $X_i$, each containing 10 elements, the total “membership count” (the sum of the sizes of all the $X_i$) is:
$$
50 \times 10 = 500.
$$
Step 3: Relate “Membership Count” to the Exact 20 Memberships per Element
Each element of $T$ is in exactly 20 of the $X_i$. Therefore, if $|T|$ is the number of distinct elements in $T$, the total membership count across all $X_i$ is also:
$$
|T| \times 20.
$$
Equating this to the membership count we found in Step 2 gives:
$$
|T| \times 20 = 500 \quad \Rightarrow \quad |T| = \frac{500}{20} = 25.
$$
Step 4: Compute the Total “Membership Count” for $Y_j$ Sets
We have $n$ sets $Y_j$, each containing 5 elements, so the total “membership count” is:
$$
n \times 5.
$$
Step 5: Relate “Membership Count” to the Exact 6 Memberships per Element
Each element of $T$ is in exactly 6 of the $Y_j$. Since $|T| = 25$, the total membership count across all $Y_j$ is:
$$
25 \times 6 = 150.
$$
Equate this to the membership count for the $Y_j$:
$$
n \times 5 = 150 \quad \Rightarrow \quad n = \frac{150}{5} = 30.
$$
Step 6: Conclusion
Hence, the required value of $n$ is:
$$
\boxed{30}.
$$
