If the coefficients of three consecutive terms in the expansion of `(1 + x)^(n)` are 165,330 and 462 respectively , the value of n is is

Options

A. 7
B. 9
C. 11
D. 13

Answer

To find the value of \( n \) given the coefficients of three consecutive terms in the expansion of \( (1 + x)^n \) are 165, 330, and 462, we can follow these steps:

### Step 1: Identify the coefficients
Let the three consecutive terms be represented as:
- \( T_r = \binom{n}{r} \)
- \( T_{r+1} = \binom{n}{r+1} \)
- \( T_{r+2} = \binom{n}{r+2} \)

From the problem, we have:
\[ \binom{n}{r} = 165, \quad \binom{n}{r+1} = 330, \quad \binom{n}{r+2} = 462 \]

### Step 2: Set up the equations
Using the properties of binomial coefficients, we can express the relationships between these coefficients:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{330}{165} = 2 \]
\[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{462}{330} = \frac{7}{5} \]

### Step 3: Simplify the first ratio
From the first ratio:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{n - r}{r + 1} = 2 \]
This leads to:
\[ n - r = 2(r + 1) \implies n - r = 2r + 2 \implies n = 3r + 2 \quad \text{(Equation 1)} \]

### Step 4: Simplify the second ratio
From the second ratio:
\[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{n - r - 1}{r + 2} = \frac{7}{5} \]
This leads to:
\[ 5(n - r - 1) = 7(r + 2) \]
Expanding this gives:
\[ 5n - 5r - 5 = 7r + 14 \implies 5n - 12r = 19 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 1 into Equation 2
Substituting \( n = 3r + 2 \) into Equation 2:
\[ 5(3r + 2) - 12r = 19 \]
Expanding this gives:
\[ 15r + 10 - 12r = 19 \implies 3r + 10 = 19 \implies 3r = 9 \implies r = 3 \]

### Step 6: Find \( n \)
Substituting \( r = 3 \) back into Equation 1:
\[ n = 3(3) + 2 = 9 + 2 = 11 \]

### Conclusion
Thus, the value of \( n \) is \( \boxed{11} \).

---