Question

The value of
$(\frac{30}{0})(\frac{30}{10})-(\frac{30}{1})(\frac{30}{11})+(\frac{30}{2})(\frac{30}{12})...+(\frac{30}{20})(\frac{30}{30})$ is; where $(\frac{n}{r}){=}^n{C}_r$

• A. ${(}_{10}^{30})$
• B. ${(}_{15}^{30})$
• C. ${(}_{30}^{60})$
• D. ${(}_{10}^{31})$

Answer 1

The correct option is A ${(}_{10}^{30})$

To find
${}^{30}{C}_0{}^{30}{C}_{10}{-}^{30}{C}_1^{30}{C}_{11}{+}^{30}{C}_2^{30}{C}_{12}-...{+}^{30}{C}_{20}{}^{30}{C}_{30}$
We know that
$(1+x{)}^{30}={}^{30}{C}_0{+}^{30}{C}_{1}x{+}^{30}{C}_2{x}^2+.....{+}^{30}{C}_{20}{x}^{20}+..{.}^{30}{C}_{30}{x}^{30}.....(1\left)\right(x-1{)}^{30}={}^{30}{C}_0{x}^{30}{-}^{30}{C}_1{x}^{29}+....{+}^{30}{C}_{10}{x}^{20}{-}^{30}{C}_{11}{x}^{19}{+}^{30}{C}_{12}{x}^{18}+..{.}^{30}{C}_{30}{x}^0.....\left(2\right)$
Multiplying equation (1) and (2),we get
$(x^2-1{)}^{30}=()\times ()$
Equating the coefficients of $x^{20}$ on both sides, we get
${}^{30}{C}_{10}{=}^{30}{C}_0^{30}{C}_{10}{-}^{30}{C}_1^{30}{C}_{11}{+}^{30}{C}_2^{30}{C}_{12}-...{+}^{30}{C}_{20}{}^{30}{C}_{30}$
$\therefore$ Req.value is ${}^{30}{C}_{10}$