Question

The total number of distinct terms in the expansion of ${(x+a)}^{100}+{(x-a)}^{100}$ after simplification is

• A. $50$
• B. $202$
• C. $51$
• D. $noneofthese$

Answer 1

The correct option is C $51$

Number of terms in a binomial expansion is$=(n+1)$

When n is even,in the expansion${(x+a)}^n+{(x-a)}^n=2\{{{}^{n}C}_0{x}^n+{{}^{n}C}_2{x}^{n-2}{a}^2+...+a^n\}$

Thus,the odd number of terms get cancelled and even number of terms get added except the first term.

Therefore,total number of terms is=$(\frac{n}{2}+1)$terms

Hence,${(x+a)}^{100}+{(x-a)}^{100}$ has

$(\frac{100}{2}+1)terms=51$ terms therefore Number of terms$=51$terms