Question:
The total number of terms in the expansion of $(x+a)^{100}+(x-a)^{100}$ after simplification is
(A) 50
(B) 202
(C) 51
(D) none of these
Solution:
(C) 51
Explanation:
Given $(x+a)^{100}+(x-a)^{100}$
$=\left({ }^{100} C_{0} x^{100}+{ }^{100} C_{1} x^{99} a+{ }^{100} C_{2} x^{98} a^{2}+\ldots\right)$
$+\left({ }^{100} C_{0} x^{100}-{ }^{100} C_{1} x^{99} a+{ }^{100} C_{2} x^{98} a^{2}+\ldots\right)$
$=2\left({ }^{100} C_{0} x^{100}+{ }^{100} C_{2} x^{98} a^{2}+\ldots++{ }^{100} C_{100} a^{100}\right)$
So, there are 51 terms
Hence option $c$ is the correct answer.