Question

The sum$\sum _{i=0}^m{}^{10}C_i\times {}^{20}C_{m-i}$ (where ${}^p{C}_q=0,$if $p<q$ ) is maximum when$m$ is:

• A. $5$
• B. $15$
• C. $10$
• D. $20$

Answer 1

The correct option is B

$15$

Explanation for the correct option:

Find the required value.

Given:$\sum _{i=0}^m{}^{10}C_i\times {}^{20}C_{m-i}$

If we expand the above expression we get,

${}^{10}C_0{}^{20}C_m+{}^{10}C_1{}^{20}C_{m-1}+{}^{10}C_2{}^{20}C_{m-2}+.....+{}^{10}C_m{}^{20}C_0$

We can analyses this as, $10$ red balls &$20$ blue balls are kept and we have to choose $m$ number of balls.

$\therefore$ Sum will be maximum when $m=\frac{\mathrm{no}.\mathrm{of}\mathrm{red}\mathrm{balls}+\mathrm{no}.\mathrm{of}\mathrm{blue}\mathrm{balls}}{2}$

$$\begin{gathered} =\frac{10+20}{2} \\ =15 \end{gathered}$$

Hence, option (B) is the correct answer.