In the expansion of `(1+x)^(n)` the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, then the value of n =

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem, we need to find the value of \( n \) given the binomial coefficients of three consecutive terms in the expansion of \( (1 + x)^n \) are 220, 495, and 792.

### Step-by-step Solution:

1. **Understanding Binomial Coefficients**:
The binomial coefficient for the \( r+1 \)th term in the expansion of \( (1 + x)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} \]
Therefore, we can denote the three consecutive terms as:
\[ \binom{n}{r-1} = 220, \quad \binom{n}{r} = 495, \quad \binom{n}{r+1} = 792 \]

2. **Using the Property of Binomial Coefficients**:
We can use the relationship between consecutive binomial coefficients:
\[ \binom{n}{r} = \frac{n - r + 1}{r} \cdot \binom{n}{r-1} \]
This gives us:
\[ 495 = \frac{n - r + 1}{r} \cdot 220 \]

3. **Setting Up the First Equation**:
Rearranging the equation:
\[ \frac{n - r + 1}{r} = \frac{495}{220} = \frac{9}{4} \]
Cross-multiplying gives:
\[ 4(n - r + 1) = 9r \]
Simplifying this, we get:
\[ 4n - 4r + 4 = 9r \implies 4n - 13r + 4 = 0 \quad \text{(Equation 1)} \]

4. **Setting Up the Second Equation**:
Now using the next pair of coefficients:
\[ \binom{n}{r+1} = \frac{n - r}{r + 1} \cdot \binom{n}{r} \]
This gives us:
\[ 792 = \frac{n - r}{r + 1} \cdot 495 \]
Rearranging gives:
\[ \frac{n - r}{r + 1} = \frac{792}{495} = \frac{8}{5} \]
Cross-multiplying gives:
\[ 5(n - r) = 8(r + 1) \]
Simplifying this, we get:
\[ 5n - 5r = 8r + 8 \implies 5n - 13r - 8 = 0 \quad \text{(Equation 2)} \]

5. **Solving the System of Equations**:
Now we have two equations:
- \( 4n - 13r + 4 = 0 \) (Equation 1)
- \( 5n - 13r - 8 = 0 \) (Equation 2)

We can eliminate \( r \) by subtracting Equation 1 from Equation 2:
\[ (5n - 13r - 8) - (4n - 13r + 4) = 0 \]
This simplifies to:
\[ n - 12 = 0 \implies n = 12 \]

### Final Answer:
The value of \( n \) is \( \boxed{12} \).