The sum of the binomial coefficients of the 3rd, 4th terms from the beginning and from the end of `(a+ x)^n` is 440 then n =

Options

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

Answer

To solve the problem step by step, we need to find the value of \( n \) given that the sum of the binomial coefficients of the 3rd and 4th terms from both the beginning and the end of the expansion of \( (a + x)^n \) is 440.

### Step 1: Identify the Binomial Coefficients
The coefficients of the terms in the binomial expansion can be expressed as follows:
- The coefficient of the 3rd term from the beginning is \( \binom{n}{2} \).
- The coefficient of the 4th term from the beginning is \( \binom{n}{3} \).
- The coefficient of the 3rd term from the end is \( \binom{n}{n-2} = \binom{n}{2} \).
- The coefficient of the 4th term from the end is \( \binom{n}{n-3} = \binom{n}{3} \).

### Step 2: Set Up the Equation
According to the problem, the sum of these coefficients is given as:
\[ \binom{n}{2} + \binom{n}{3} + \binom{n}{2} + \binom{n}{3} = 440 \]
This simplifies to:
\[ 2\binom{n}{2} + 2\binom{n}{3} = 440 \]
Dividing the entire equation by 2 gives:
\[ \binom{n}{2} + \binom{n}{3} = 220 \]

### Step 3: Express the Binomial Coefficients
Using the formula for binomial coefficients:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
\[ \binom{n}{3} = \frac{n(n-1)(n-2)}{6} \]
Substituting these into our equation:
\[ \frac{n(n-1)}{2} + \frac{n(n-1)(n-2)}{6} = 220 \]

### Step 4: Find a Common Denominator
The common denominator for the fractions is 6. Therefore, we can rewrite the equation as:
\[ \frac{3n(n-1)}{6} + \frac{n(n-1)(n-2)}{6} = 220 \]
Combining the fractions gives:
\[ \frac{3n(n-1) + n(n-1)(n-2)}{6} = 220 \]

### Step 5: Multiply Through by 6
To eliminate the fraction, multiply both sides by 6:
\[ 3n(n-1) + n(n-1)(n-2) = 1320 \]

### Step 6: Factor Out Common Terms
Factoring out \( n(n-1) \):
\[ n(n-1)(3 + (n-2)) = 1320 \]
This simplifies to:
\[ n(n-1)(n + 1) = 1320 \]

### Step 7: Solve for \( n \)
Now we need to find integer values of \( n \) that satisfy this equation. We can test possible values of \( n \) (10, 11, 12, 13) to find the correct one.

1. For \( n = 10 \):
\[ 10(9)(11) = 990 \quad (\text{not a solution}) \]
2. For \( n = 11 \):
\[ 11(10)(12) = 1320 \quad (\text{this works!}) \]
3. For \( n = 12 \):
\[ 12(11)(13) = 1716 \quad (\text{not a solution}) \]
4. For \( n = 13 \):
\[ 13(12)(14) = 2184 \quad (\text{not a solution}) \]

### Conclusion
The only value of \( n \) that satisfies the equation is \( n = 11 \).

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