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:
Show Hint
- 2196
- 2172
- 2184
- 2148
The Correct Option is C
Solution and Explanation
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).
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