Suppose `x_1,x_2,….x_(50)` are 50 sets each having 10 elements and `Y_1,Y_2,….Y_n` are n sets each having 5 elements. Let `uu_(i=1)^50 X_i=uu_(i=1)^n Y_i=Z` and each element of Z belong to exactly 25 of `X_i` and exactly 6 of `Y_i` then value of n is

Options

A. 20
B. 22
C. 24
D. 26

Answer

To solve the problem, we need to analyze the information given about the sets and the elements they contain.

### Step-by-Step Solution:

1. **Understanding the Sets**:
- We have 50 sets \( X_1, X_2, \ldots, X_{50} \), each containing 10 elements.
- Therefore, the total number of elements in all \( X_i \) is:
\[ \text{Total elements in } X = 50 \times 10 = 500. \]

2. **Union of Sets**:
- Let \( Z \) be the union of all the sets \( X_i \) and \( Y_i \).
- Each element of \( Z \) belongs to exactly 25 of the \( X_i \) sets.

3. **Counting Contributions from \( X_i \)**:
- If each element of \( Z \) belongs to 25 sets, then the total contribution to the union from all elements in \( Z \) is:
\[ \text{Total contributions from } X = \frac{500}{25} = 20. \]
- This means there are 20 distinct elements in \( Z \).

4. **Analyzing the Sets \( Y_i \)**:
- Let \( n \) be the number of sets \( Y_1, Y_2, \ldots, Y_n \), and each set contains 5 elements.
- Therefore, the total number of elements in all \( Y_i \) is:
\[ \text{Total elements in } Y = n \times 5. \]

5. **Counting Contributions from \( Y_i \)**:
- Each element of \( Z \) belongs to exactly 6 of the \( Y_i \) sets.
- Thus, the total contribution to the union from all elements in \( Z \) from the \( Y_i \) sets is:
\[ \text{Total contributions from } Y = \frac{n \times 5}{6}. \]

6. **Setting Up the Equation**:
- Since the total contributions from both sets must equal the number of distinct elements in \( Z \), we can set up the equation:
\[ 20 = \frac{n \times 5}{6}. \]

7. **Solving for \( n \)**:
- To solve for \( n \), we multiply both sides by 6:
\[ 120 = n \times 5. \]
- Now, divide both sides by 5:
\[ n = \frac{120}{5} = 24. \]

### Final Answer:
Thus, the value of \( n \) is \( \boxed{24} \).