Question

If $x+y=1$ then,${\sum }_{r=0}^n{}^n{C}_r{x}^r{y}^{n-r}$ equals

• A. $1$
• B. $n$
• C. $nx$
• D. $ny$

Answer 1

Answer: (A)

$1$

Using binomial expansion

$(x+y{)}^n={}^n{C}_0{x}^0{y}^{n-0}+{}^n{C}_1{x}^1{y}^{n-1}+{}^n{C}_2{x}^2{y}^{n-2}+...\dots ..+{}^n{C}_n{x}^n{y}^{n-n}$

$=\sum _{r=0}^n{}^n{C}_r{x}^r{y}^{n-r}$

at $x+y=1$

$1^n=\sum _{r=0}^n{}^n{C}_r\times x^r{y}^{n-r}=1$.