If `""^(n)C_(7)= ""^(n)C_(5)`, then `""C_(4)=` ………….

Answer

To solve the equation \( \binom{n}{7} = \binom{n}{5} \), we can use the property of combinations that states:

\[ \binom{n}{r} = \binom{n}{n-r} \]

### Step 1: Apply the property of combinations

Given that \( \binom{n}{7} = \binom{n}{5} \), we can set \( r = 7 \) and \( n - r = 5 \). This implies:

\[ n - 7 = 5 \]

### Step 2: Solve for \( n \)

Now, we can solve for \( n \):

\[ n = 5 + 7 = 12 \]

### Step 3: Find \( \binom{n}{4} \)

Now that we have \( n = 12 \), we need to find \( \binom{12}{4} \):

\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

### Step 4: Simplify \( \binom{12}{4} \)

Calculating \( \binom{12}{4} \):

\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]

Calculating the denominator:

\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]

Now, divide the numerator by the denominator:

\[ \binom{12}{4} = \frac{11880}{24} = 495 \]

### Final Answer

Thus, \( \binom{12}{4} = 495 \).

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