If `""^(n)C_(10) = ""^(n) C _ (14),` find the value of ` ""^(n) C_(20) and ""^(25)C_n`.

Answer

To solve the problem, we need to find the value of \( n \) given that \( \binom{n}{10} = \binom{n}{14} \), and then use this value to find \( \binom{n}{20} \) and \( \binom{25}{n} \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We start with the equation:
\[ \binom{n}{10} = \binom{n}{14} \]
This means that the number of ways to choose 10 items from \( n \) is equal to the number of ways to choose 14 items from \( n \).

2. **Using the Combination Formula**:
Recall the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Applying this to our equation:
\[ \frac{n!}{10!(n-10)!} = \frac{n!}{14!(n-14)!} \]

3. **Canceling \( n! \)**:
Since \( n! \) is common on both sides, we can cancel it:
\[ \frac{1}{10!(n-10)!} = \frac{1}{14!(n-14)!} \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 14!(n-14)! = 10!(n-10)! \]

5. **Expanding Factorials**:
We can expand \( 14! \) and \( 10! \):
\[ 14 \times 13 \times 12 \times 11 \times 10! \times (n-14)! = 10! \times (n-10)(n-11)(n-12)(n-13)(n-14)! \]
Canceling \( 10! \) and \( (n-14)! \) from both sides gives:
\[ 14 \times 13 \times 12 \times 11 = (n-10)(n-11)(n-12)(n-13) \]

6. **Finding \( n \)**:
We can set \( n - 10 = x \), so:
\[ 14 \times 13 \times 12 \times 11 = x(x-1)(x-2)(x-3) \]
We know \( 14 \times 13 \times 12 \times 11 = 24024 \). We need to find \( x \) such that:
\[ x(x-1)(x-2)(x-3) = 24024 \]
Testing \( x = 14 \):
\[ 14 \times 13 \times 12 \times 11 = 24024 \]
Thus, \( x = 14 \) implies \( n - 10 = 14 \) or \( n = 24 \).

7. **Calculating \( \binom{n}{20} \)**:
Now we calculate \( \binom{24}{20} \):
\[ \binom{24}{20} = \binom{24}{4} = \frac{24!}{4!(24-4)!} = \frac{24 \times 23 \times 22 \times 21}{4 \times 3 \times 2 \times 1} \]
Calculating this gives:
\[ = \frac{24 \times 23 \times 22 \times 21}{24} = 23 \times 22 \times 21 = 10626 \]

8. **Calculating \( \binom{25}{n} \)**:
Now we calculate \( \binom{25}{24} \):
\[ \binom{25}{24} = 25 \]

### Final Answers:
- \( \binom{24}{20} = 10626 \)
- \( \binom{25}{n} = 25 \)