The number of terms in the expansion of `(x+a)^100+(x-a)^100` after simplification

Options

A. `50`
B. `202`
C. `51`
D. None of these

Answer

C

Explanation

Here, `(x + a)^(100) + (x - a)^(100)`
total number of terms is 102 in the expansion of `(x + a)^(100) + (x - a)^(100)`
50 terms of `(x + a)^(100)` cancel out 50 terms of `(x - a)^(100)` 51 terms of `(x + a)^(100)` get added to the 51 terms of `(x - a)^(100)`.
Alternate Method
`(x + a)^(100) + (x - a)^(100) = .^(100)C_(0) x^(100) + .^(100)C_(1) x^(99) a+ .... + .^(100)C_(100) a^(100)`
`+ .^(100)C_(0) x^(100) - .^(100)C_(1) x^(99) a + .... + .^(100)C_(100) a^(100)`
`= 2ubrace([.^(100)C_(0) x^(100) + .^(100)C_(2) x^(98) a^(2) + ...+ .^(100)C_(100)a^(100)])_(51" terms")`