Question

Value of ${20}_1^C-2\times {20}_2^C+3\times {20}_3^C$...$20\times {20}_2^{C}0$ is

• A. $2^{19}$
• B. $0$
• C. $2^{20}-1$
• D. none of these

Answer 1

The correct option is B $0$

Consider
$(1-x{)}^{20}$
$=1-{}^{20}{C}_1{x}^1+{}^{20}{C}_2{x}^2-{}^{20}{C}_3{x}^3...+{}^{20}{C}_{20}{x}^{20}$
By differentiating with respect to x, we get
$-n(1-x{)}^{n-1}=-{}^{20}{C}_1+2{}^{20}{C}_2{x}^1-3{}^{20}{C}_3{x}^2+...20{}^{20}{C}_{20}{x}^{19}$
Substituting x=1, we get
$0=-{}^{20}{C}_1+2{}^{20}{C}_2-3{}^{20}{C}_3+...20{}^{20}{C}_{20}$
Hence answer is $0$.