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: The number of integral terms in the binomial expansion \( \left( \sqrt{7} + \sqrt{11} \right)^n \) can be found by considering the terms of the form \( \binom{n}{k} \sqrt{7}^{n-k} \sqrt{11}^k \). For an integral term, the exponents of both square roots must be even. 
Step 2: The number of integral terms is the number of valid values of \( k \) such that both \( n-k \) and \( k \) are even. This means \( k \) must range from 0 to \( n \), and \( k \) must be even. 
Step 3: Solve the equation \( \frac{n}{2} + 1 = 183 \), which gives \( n = 2184 \). Thus, the correct answer is (3).