The total number of terms in the expansion of (x+a)^100+(x-a)^100 is

the expression is:

therefore the number of terms will be 51. 

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(x+a)^100 would have 101 terms x^100 + a*x^99 ... + a^100
(x-a)^100 would have 101 terms x^100 - a*x^99 ...  + a^100
when added all the odd exponents of x would get cancelled
so we would remain with 51 terms
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