Question

The sum ${\sum }_{i=0}^m{(}_i^{10}\left){(}_{m-i}^{20}\right)$, (where ${(}_q^p)$ = 0 if $p<q)$ is maximum when $m$ is

• A. 5
• B. 10
• C. 15
• D. 20

Answer 1

The correct option is C 15

${\sum }_{i=0}^m{}^{10}{C}_i{}^{20}{C}_{m-i}={}^{10}{C}_0{}^{20}{C}_m+{}^{10}{C}_1{}^{20}{C}_{m-1}+{}^{10}{C}_2{}^{20}{C}_{m-2}+.....+{}^{10}{C}_m{}^{20}{C}_0$
= Coeff of $x^m$ in the expansion of product $(1+x{)}^{10}(1+x{)}^{20}$
= Coeff of $x^m$ in the expansion of $(1+x{)}^{30}$
= ${}^{30}{C}_m$
To get max. value of given sum ${}^{30}{C}_m$ should be max. which is so when $m=\frac{30}{2}=15$.
$\left[\text{Using the fact that max}{(}^n{C}_r\right)={}^n{C}_{n/2};\text{if}n{\text{is even OR}}^n{C}_{\frac{n+1}{2}};\text{if}n\text{is odd}]$