First slide

Combinations

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

Moderate

A

5
B

10
C

15
D

20

Solution

We have,
$\sum_{i = 0}^{m}^{10} C_{i} \times^{20} C_{m - i}$
= Number of ways of selecting $m$ objects from two groups of objects one group containing 10 distinct objects of one type and another group containing 20 distinct objects of second kind.
$=^{30} C_{m}$
Clearly, $^{30} C_{m}$ is maximum when $m = 15$ , because $^{n} C_{r}$ is
maximum for $r = n / 2$ if $n$ is even and when $n$ is odd it is
maximum for $r = \frac{n + 1}{2} or \frac{n - 1}{2}$ .

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