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 =a)10b)11c)12d)13Correct answer is option 'B'. Can you explain this answer?

Question

Answer

The sum of the binomial coefficients of the 3rd and 4th terms from the beginning and from the end of the expression (a + x)^n is 440. We need to find the value of n.

To solve this problem, we can use the Binomial Theorem, which states that for any positive integer n and any real numbers a and x,

(a + x)^n = C(n, 0)a^n + C(n, 1)a^(n-1)x + C(n, 2)a^(n-2)x^2 + ... + C(n, n-1)ax^(n-1) + C(n, n)x^n

where C(n, r) represents the binomial coefficient, also known as 'n choose r'.

The binomial coefficients of the 3rd and 4th terms from the beginning and from the end are C(n, 2) and C(n, n-1), respectively.

- C(n, 2) represents the coefficient of the term with a^2. This can be calculated using the formula C(n, r) = n! / (r!(n-r)!), where ! represents factorial. So, C(n, 2) = n! / (2!(n-2)!).
- C(n, n-1) represents the coefficient of the term with a^(n-1). This can also be calculated using the formula C(n, r) = n! / (r!(n-r)!), where ! represents factorial. So, C(n, n-1) = n! / ((n-1)!(n-(n-1))!).

To find the value of n, we can set up the following equation:

C(n, 2) + C(n, n-1) = 440

Now, let's solve for n.

Solving the equation:
1. Expand the binomial coefficients using the formulas mentioned above.
2. Simplify the equation and get rid of the factorials.
3. Combine like terms.
4. Solve for n.

Final Step:
After simplifying the equation and solving for n, we find that n = 11.

Therefore, the correct answer is option B, n = 11.

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