Question

The value of
$\binom{30}{0}\binom{30}{10} -\binom{30}{1}\binom{30}{11}+\binom{30}{2}\binom{30}{12}+....+\binom{30}{20}\binom{30}{30} =$

• A. ${ }^{60} C _{20}$
• B. ${ }^{30} C _{10}$
• C. ${ }^{60} C _{30}$
• D. ${ }^{40} C _{30}$

Answer 1

Answer: (B)

$(1-x)^{30}={ }^{30} C_{0}\, x^{0}-{ }^{30} C_{1} \,x^{1}+{ }^{30} C_{2} \,x^{2}+\ldots .+(-1)^{30}{ }^{30} C_{30} \,x^{30} \,\,\,...(i)$
$(x+1)^{30}={ }^{30} C_{0} \,x^{30}+{ }^{30} C_{1} \,x^{29}+{ }^{30} C_{2} \,x^{28}+\ldots .+{ }^{30} C_{10} x^{20} +\ldots .+{ }^{30} C_{30} \,x^{0} \,\,\,...(ii)$
Multiplying (i) and (ii) and equating the coefficient of $x^{20}$ on both sides, we get required sum is equal to coefficient of $x ^{20}$ in $\left(1- x ^{2}\right)^{30}$, which is given by ${ }^{30} C _{10}$.