Question

The value of ${}^{10}{C}_1+{}^{10}{C}_2+{}^{10}{C}_3+...+{}^{10}{C}_9$ is

• A. $2^{10}$
• B. $2^{11}$
• C. $2^{10}-2$
• D. $2^{10}-1$

Answer 1

Answer: (C)

$2^{10}-2$

Consider $(1+x{)}^{10}={}^{10}{C}_0{x}^{10}+{}^{10}{C}_1{x}^9+{}^{10}{C}_2{x}^8+.........+{}^{10}{C}_{9}x+{}^{10}{C}_{10}$

Now substitute $x=1$ both sides

$(1+1{)}^{10}={}^{10}{C}_0+{}^{10}{C}_1+{}^{10}{C}_2+.........+{}^{10}{C}_9+{}^{10}{C}_{10}$

$\Rightarrow 2^{10}=1+{}^{10}{C}_1+{}^{10}{C}_2+.........+{}^{10}{C}_9+1$

$\therefore {}^{10}{C}_1+{}^{10}{C}_2+{}^{10}{C}_3+......+{}^{10}{C}_9=2{}^{10}-2$

Note: ${}^n{C}_0=1$ and ${}^n{C}_n=1$