Question

The value of ${}^{30}{C}_0{}^{30}{C}_{10}{-}^{30}{C}_1{}^{30}{C}_{11}{+}^{30}{C}_2{}^{30}{C}_{12}.....{+}^{30}{C}_{10}{}^{30}{C}_{20}$ is:

• A. ${}^{30}{C}_{10}$
• B. ${}^{30}{C}_{15}$
• C. ${}^{30}{C}_{20}$
• D. ${}^{60}{C}_{30}$

Answer 1

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

There is a slight error in the question, the last term should be ${}^{30}{C}_{10}{.}^{30}{C}_{20}$ instead of ${}^{30}{C}_{20}{.}^{30}{C}_{20}$

Now, ${(1+x)}^{30}{=}^{30}{C}_0{+}^{30}{C}_1.x{+}^{30}{C}_2.x^2+...{+}^{30}{C}_{30}.x^{30}$

and, ${(1+x)}^{30}{=}^{30}{C}_0{-}^{30}{C}_1.x{+}^{30}{C}_2.x^2-...{+}^{30}{C}_{30}.x^{30}$

Hence, the value of ${}^{30}{C_0}^{30}{C}_{10}{-}^{30}{C_1}^{30}{C}_{11}{+}^{30}{C_2}^{30}{C}_{12}-....{+}^{30}{C_{10}}^{30}{C}_{20}=$

co-efficient of $x^{30}$ in ${(1+x)}^{30}{(1-x)}^{30}=$ co-efficient of $x^{30}$ in $(1-x^2{)}^{30}{=}^{30}{C}_r{x}^{2r}$

So, the coefficients are: ${}^{30}{C}_{10}$ and ${}^{30}{C}_{20}$ as both are equal.

Hence, $A$ and $C$ are correct.