Question

The total number of terms in the expansion of $(x+y{)}^{50}+(x-y{)}^{50}$ is

• A. $51$
• B. $26$
• C. $102$
• D. $25$

Answer 1

Answer: (B)

Consider given the expression,

${(x+y)}^{50}+{(x-y)}^{50}$

We know that,

${(x+y)}^n={}^n{C}_0{x}^n{+}^n{C}_1{x}^{n-1}y{+}^n{C}_2{x}^{n-2}{y}^2+{......}^n{C}_r{x}^{n-r}{y}^r{.....}^n{C}_n{y}^n$

Now,

${(x+y)}^{50}+{(x-y)}^{50}$

$={}^{50}{C}_0{x}^{50}{+}^{50}{C}_1{x}^{49}y{+}^{50}{C}_2{x}^{48}{y}^2+{.....}^{50}{C}_{50}{y}^{50}+{}^{50}{C}_0{x}^{50}{-}^{50}{C}_1{x}^{49}y{+}^{50}{C}_2{x}^{48}{y}^2+.....{+}^{50}{C}_{50}{y}^{50}$

$=2{[}^{50}{C}_0{x}^{50}{+}^{50}{C}_2{x}^{50-2}y{+}^{50}{C}_4{x}^{50-4}{y}^4+{......}^{50}{C}_r{x}^{50-r}{y}^r{.....}^{50}{C}_{50}{y}^{50}]$

Here, ${}^{50}{C}_0,{}^{50}{C}_1,{}^{50}{C}_2,...{}^{50}{C}_{49},{}^{50}{C}_{50}$ contain $51$ terms.

In $26$-coefficients , $r$ is even.

In $25$-coefficients, $r$ is odd.

Therefore, there are $26$ terms

Hence, Option B is correct.