If `""^(22)C_(r)` is the largest binomial coefficient in the expansion of `(1+x)^(22)`
, find the value of `""^(13)C_(r)`.

Answer

To solve the problem, we need to find the value of \( \binom{13}{r} \) given that \( \binom{22}{r} \) is the largest binomial coefficient in the expansion of \( (1+x)^{22} \).

### Step-by-Step Solution:

1. **Identify the largest binomial coefficient**:
In the expansion of \( (1+x)^{n} \), the largest binomial coefficient occurs at \( r = \frac{n}{2} \) when \( n \) is even. Here, \( n = 22 \), which is even.

\[ r = \frac{22}{2} = 11 \]

2. **Equate the given condition**:
Since it is given that \( \binom{22}{r} \) is the largest coefficient, we can set:

\[ r = 11 \]

3. **Find \( \binom{13}{r} \)**:
Now we need to calculate \( \binom{13}{r} \) where \( r = 11 \).

\[ \binom{13}{11} = \binom{13}{2} \]

(Using the property \( \binom{n}{r} = \binom{n}{n-r} \))

4. **Calculate \( \binom{13}{2} \)**:
The formula for binomial coefficients is:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For \( \binom{13}{2} \):

\[ \binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = \frac{156}{2} = 78 \]

5. **Final Answer**:
Therefore, the value of \( \binom{13}{r} \) is:

\[ \boxed{78} \]