If `.^(n^(2)-.^(n))C_(2)=.^(n^(2)-n)C_(4)` , then n is equal to

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the equation \( C(n^2 - n, 2) = C(n^2 - n, 4) \), we can follow these steps:

### Step 1: Write down the combinations
We know that the combination formula is given by:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Thus, we can express the combinations as:
\[ C(n^2 - n, 2) = \frac{(n^2 - n)!}{2!(n^2 - n - 2)!} \]
and
\[ C(n^2 - n, 4) = \frac{(n^2 - n)!}{4!(n^2 - n - 4)!} \]

### Step 2: Set the combinations equal to each other
From the problem, we have:
\[ \frac{(n^2 - n)!}{2!(n^2 - n - 2)!} = \frac{(n^2 - n)!}{4!(n^2 - n - 4)!} \]

### Step 3: Cancel out the common factorials
Since \( (n^2 - n)! \) is present in both sides, we can cancel it out (assuming \( n^2 - n \geq 4 \)):
\[ \frac{1}{2!(n^2 - n - 2)!} = \frac{1}{4!(n^2 - n - 4)!} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 4!(n^2 - n - 2)! = 2!(n^2 - n - 4)! \]

### Step 5: Substitute the factorial values
We know that \( 4! = 24 \) and \( 2! = 2 \):
\[ 24(n^2 - n - 2)! = 2(n^2 - n - 4)! \]

### Step 6: Simplify the equation
Dividing both sides by 2 gives:
\[ 12(n^2 - n - 2)! = (n^2 - n - 4)! \]

### Step 7: Express \( (n^2 - n - 2)! \) in terms of \( (n^2 - n - 4)! \)
We can express \( (n^2 - n - 2)! \) as:
\[ (n^2 - n - 2)! = (n^2 - n - 2)(n^2 - n - 3)(n^2 - n - 4)! \]
Thus, substituting this back into the equation gives:
\[ 12(n^2 - n - 2)(n^2 - n - 3)(n^2 - n - 4)! = (n^2 - n - 4)! \]

### Step 8: Cancel \( (n^2 - n - 4)! \)
Assuming \( (n^2 - n - 4)! \neq 0 \), we can cancel it out:
\[ 12(n^2 - n - 2)(n^2 - n - 3) = 1 \]

### Step 9: Rearranging the equation
This leads us to:
\[ (n^2 - n - 2)(n^2 - n - 3) = \frac{1}{12} \]

### Step 10: Solve for \( n \)
Now we can solve the quadratic equation:
\[ n^2 - n - 2 = 0 \quad \text{and} \quad n^2 - n - 3 = 0 \]
Factoring gives us:
\[ (n - 3)(n + 2) = 0 \]
Thus, \( n = 3 \) or \( n = -2 \).

Since \( n \) must be a non-negative integer, we conclude:
\[ n = 3 \]

### Final Answer
Thus, the value of \( n \) is \( 3 \).

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