Question

The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( \left( \sqrt{7} + \sqrt{11} \right)^n \) is 183, is:

• A. 2196
• B. 2172
• C. 2184
• D. 2148

Hint:

To find the number of integral terms in a binomial expansion involving square roots, focus on the exponents and ensure both terms have even exponents. Solve for \( n \) by counting the number of such terms.

Answer 1

The Correct Option is C

Step 1: To determine the count of integer terms in the binomial expansion \( \left( \sqrt{7} + \sqrt{11} \right)^n \), examine terms of the form \( \binom{n}{k} \sqrt{7}^{n-k} \sqrt{11}^k \). For a term to be an integer, the powers of both square roots, \( n-k \) and \( k \), must be even.

Step 2: The number of integer terms corresponds to the count of values for \( k \) (where \( 0 \le k \le n \)) such that both \( n-k \) and \( k \) are even. This implies \( k \) must be an even integer.

Step 3: Solving the equation \( \frac{n}{2} + 1 = 183 \) yields \( n = 2184 \). Therefore, the correct option is (3).