If `""^n C_2: "^n C_4=2:1` then the value of n is

Options

A. 8
B. 7
C. 6
D. 5

Answer

To solve the problem \( \binom{n}{2} : \binom{n}{4} = 2 : 1 \), we can follow these steps:

### Step 1: Write the expressions for \( \binom{n}{2} \) and \( \binom{n}{4} \)

The binomial coefficient \( \binom{n}{r} \) is given by the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Thus, we can write:
\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]
and
\[ \binom{n}{4} = \frac{n!}{4!(n-4)!} = \frac{n(n-1)(n-2)(n-3)}{24} \]

### Step 2: Set up the ratio

According to the problem, we have:
\[ \frac{\binom{n}{2}}{\binom{n}{4}} = \frac{2}{1} \]

Substituting the expressions we derived:
\[ \frac{\frac{n(n-1)}{2}}{\frac{n(n-1)(n-2)(n-3)}{24}} = \frac{2}{1} \]

### Step 3: Simplify the ratio

Now, simplify the left-hand side:
\[ \frac{n(n-1)}{2} \times \frac{24}{n(n-1)(n-2)(n-3)} = \frac{12}{(n-2)(n-3)} \]

So we have:
\[ \frac{12}{(n-2)(n-3)} = 2 \]

### Step 4: Cross-multiply to solve for \( n \)

Cross-multiplying gives:
\[ 12 = 2(n-2)(n-3) \]
Expanding the right-hand side:
\[ 12 = 2(n^2 - 5n + 6) \]
Dividing both sides by 2:
\[ 6 = n^2 - 5n + 6 \]

### Step 5: Rearranging the equation

Rearranging gives us:
\[ n^2 - 5n + 6 - 6 = 0 \implies n^2 - 5n = 0 \]

### Step 6: Factor the quadratic equation

Factoring out \( n \):
\[ n(n - 5) = 0 \]

### Step 7: Solve for \( n \)

Setting each factor to zero gives:
\[ n = 0 \quad \text{or} \quad n = 5 \]

Since \( n \) must be a positive integer (as it represents the number of items), we discard \( n = 0 \).

Thus, the value of \( n \) is:
\[ \boxed{5} \]