Total number of terms in the expansion of $(x + a)^{100} + (x - a)^{100}$ after simplification:

50

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202

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51

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None of these

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Solution

The correct option is A $51$
$(x + a)^{100} + (x - a)^{100}$
Applying binomial theorem for expansion of the following
$(x + a)^{100} + (x - a)^{100}$
$= x^{100} + x^{99} (^{100} C_{1} a) + x^{98} (^{100} C_{2} a^{2}) . . . (^{100} C_{100} a^{100})$
$+ x^{100} - x^{99} (^{100} C_{1} a) + x^{98} (^{100} C_{2} a^{2}) . . . (^{100} C_{100} a^{100})$
$= 2 [x^{100} + x^{98} (^{100} C_{2} a^{2}) + x^{96} (^{100} C_{4} a^{4}) . . . (^{100} C_{100} a^{100})]$
The powers of $a$ are in A.P
$a^{0}, a^{2}, a^{4}, a^{6} . . . a^{100}$
$0, 2, 4, 6...100$
Hence, including the term $a^{0} o r x^{100}$ there are $51$ terms.

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