Question

${}^{10}{C}_0{}^{20}{C}_{10}-{}^{10}{C}_1{}^{18}{C}_{10}+{}^{10}{C}_2{}^{16}{C}_{10}-\cdots =$

• A. $0$
• B. $1$
• C. $2^{10}$
• D. $-2^{10}$

Answer 1

Answer: (C)

${}^{10}{C}_0{}^{20}{C}_{10}-{}^{10}{C}_1{}^{18}{C}_{10}+{}^{10}{C}_2{}^{16}{C}_{10}-\cdots$
$=$ coefficient of $x^{10}$ in $[{}^{10}{C}_0(1+x{)}^{20}-{}^{10}{C}_1(1+x{)}^{18}<br/>+{}^{10}{C}_2(1+x{)}^{16}-\cdots ]$
$=$ coefficient of $x^{10}$ in $[{}^{10}{C}_0{\left((1+x{)}^2\right)}^{10}-{}^{10}{C}_1{\left((1+x{)}^2\right)}^9+{}^{10}{C}_2{\left((1+x{)}^2\right)}^8-\cdots ]$
$=$ coefficient of $x^{10}$ in ${\left[(1+x{)}^2-1\right]}^{10}$
$=$ coefficient of $x^{10}$ in ${\left[2x+x^2\right]}^{10}$
$=2^{10}$