The sum i = 0 m ( 10 i ) ( 20 m - i ) , (where ( p q ) = 0 if p q ) is maximum when m is [2002]

Question

The sum $\sum_{i = 0}^{m} (\binom{10}{i}) (\binom{20}{m - i}),$ (where $(\binom{p}{q}) = 0$ if $p > q$ ) is maximum when $m$ is [2002]

Smart Achievers · Accepted Answer

(3)

$\sum_{i = 0}^{m} {}^{10}C_{i} {}^{20}C_{m - i} = {}^{10}C_{0} {}^{20}C_{m} + {}^{10}C_{1} {}^{20}C_{m - 1} + {}^{10}C_{2} {}^{20}C_{m - 2} + \dots + {}^{10}C_{m} {}^{20}C_{0}$

$= Coeff. of x^{m} in the expansion of product {(1 + x)}^{10} {(1 + x)}^{20}$

$= Coeff. of x^{m} in the expansion of {(1 + x)}^{30} = {}^{30}C_{m}$

$\sum_{i = 0}^{m} {}^{10}C_{i} {}^{20}C_{m - i} will be maximum, if {}^{30}C_{m} will be maximum.$

$Clearly, {}^{30}C_{m} will be maximum when m = \frac{30}{2} = 15$ $[∵ \max \cdot ({}^{n}C_{r}) = {\begin{matrix} {}^{n}C_{\frac{n}{2}} if n is even \\ {}^{n}C_{\frac{n + 1}{2}} if n is odd \end{matrix}]$

Solution

Q.
The sum $\sum_{i = 0}^{m} (\binom{10}{i}) (\binom{20}{m - i}),$ (where $(\binom{p}{q}) = 0$ if $p > q$ ) is maximum when $m$ is [2002]

1 5

2 10

3 15

4 20

Want Us to Email you About Special Offers & Updates?