First slide

Combinations

Question

The sum $\sum_{i = 0}^{m} (\begin{matrix} 10 \\ i \end{matrix}) (\begin{matrix} 20 \\ m - i \end{matrix})$ , $(where (\begin{matrix} p \\ q \end{matrix}) = 0 if p < q)$ is maximum when $m$ is

Moderate

A

5
B

15
C

10
D

20

Solution

$\sum_{i = 0}^{m} (\begin{matrix} 10 \\ i \end{matrix}) (\begin{matrix} 20 \\ m - i \end{matrix}) = \sum_{i = 0}^{m}^{10} C_{i}^{20} C_{m - i}$
$=^{30} C_{m}$

Which is maximum when $m = \frac{n}{2}$ $= 15$

$(∵$ $^{n} C_{r}$ is maximum for $r = \frac{n}{2}$ when $n$ is odd)

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