Question

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

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

Answer 1

The correct option is B 51

$(x+a{)}^{100}+(x-a{)}^{100}$

$=\sum _{r=0}^{100}{}^{100}{C}_r{x}^{100-r}{a}^r+\sum _{r=0}^{100}{}^{100}{C}_r{x}^{100-r}(-a{)}^r$

$=\sum _{r=0}^{100}{}^{100}{C}_r{x}^{100-r}{a}^r(1+(1+(-1{)}^r)$

i.e. for $r$ even the term $1+(-1{)}^r$ vanishes

Thus, no, of terms correspond to the values of $r-$even

i.e. $r=0,2,\mathrm{4.....},100$

Thus, no, of terms $=51$