If`n_(C_12)=n_(C_13)`then find the value of the `25_(C_n)`

Answer

To solve the problem where \( nC_{12} = nC_{13} \), we can follow these steps:

### Step 1: Understand the relationship between combinations
We know that the combination formula states:
\[ nC_r = \frac{n!}{r!(n-r)!} \]
From the problem, we have:
\[ nC_{12} = nC_{13} \]

### Step 2: Use the property of combinations
Using the property of combinations, we can express \( nC_{13} \) in terms of \( nC_{12} \):
\[ nC_{13} = nC_{12} \implies nC_{12} = nC_{12} \]
This implies:
\[ nC_{12} = nC_{n-13} \]
Thus, we can set:
\[ n - 12 = 13 \]

### Step 3: Solve for \( n \)
From the equation \( n - 12 = 13 \), we can solve for \( n \):
\[ n = 13 + 12 = 25 \]

### Step 4: Find the value of \( 25C_n \)
Now we need to find \( 25C_n \) where \( n = 25 \):
\[ 25C_{25} = \frac{25!}{25!(25-25)!} = \frac{25!}{25! \cdot 0!} \]

### Step 5: Simplify the expression
Since \( 0! = 1 \), we have:
\[ 25C_{25} = \frac{25!}{25! \cdot 1} = 1 \]

### Final Answer
Thus, the value of \( 25C_n \) is:
\[ \boxed{1} \]

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