Question

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

• A. $100$
• B. $51$
• C. $201$
• D. $202$

Answer 1

Answer: (B)

$51$

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

$=100C_0{x}^{100}+100C_1{x}^{99}a+100C_2{x}^{98}{a}^2+...\dots .+100C_{100}{x}^o{a}^{100}+100C_0{x}^{100}-100C_1{x}^{99}a+100C_2{x}^{98}{a}^2-...\dots +100C_{100}{x}^0{a}^{100}$

Here all the odd powers of 'a' get cancelled.

The terms with $a^o,a^2,a^4,\dots ...a^{100}$ will be present.

There are $51$ terms.

i.e., $2k$ where $k=0,1,2,......50$

a

Hence the given expression has $51$ terms.