Question

The value of
$\binom{30}{0}\binom{30}{10}-\binom{30}{1}\binom{30}{11}+\binom{30}{2}\binom{30}{12} .....+ \binom{30}{20}\binom{30}{30}$
is where $\binom{n}{r} = ^{n}C_{r}$

• A. $\binom{30}{10}$
• B. $\binom{30}{15}$
• C. $\binom{60}{30}$
• D. $\binom{31}{10}$

Answer 1

Answer: (A)

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
$\left(1 + x\right)^{30} = ^{30}C_{0} + ^{30}C_{1}x + ^{30}C_{2}x^{2}$
$+ .... + ^{30}C_{20}x^{20} + ....^{30}C_{30}x^{30} \quad....\left(1\right)$
$\left(x - 1\right)^{30 }= ^{30}C_{0}x^{30} - ^{30}C_{1}x^{29 }+....+ ^{30}C_{10}x^{20}$
$- ^{30}C_{11}x^{19} + ^{30}C_{12}x^{18} +.... ^{30}C_{30}x^{0} \quad....\left(2\right)$
Multiplying $eq^{n} \left(1\right)$ and $\left(2\right)$ and equating the coefficients of $x^{20}$ on both sides, we get
$^{30}C_{10} = ^{30}C_{0}^{30}C_{10 }-^{ 30} C_{1} ^{ 30}C_{11} + ^{30}C_{2} ^{30}C_{12}- ....+ ^{30}C_{20} ^{30}C_{30}$
$\therefore \quad$ Req. value is $^{30}C_{10}$