First slide

Combinations

Question

The number of ways of choosing n objects out of $(3 n + 1)$ objects of which n are identical and $(2 n + 1)$ are distinct, is

Moderate

A

$2^{2 n}$
B

$2^{2 n + 1}$
C

$2^{2 n} - 1$
D

none of these

Solution

If we choose $k (0 \leq k \leq n)$ identical objects, then we must choose $(n - k)$ distinct objects.
This can be done in $^{2 n + 1} C_{n - k}$ ways. Thus, the required number of ways
$= \sum_{k = 0}^{n}^{2 n + 1} C_{n - k} =^{2 n + 1} C_{n} +^{2 n + 1} C_{n - 1} + \dots +^{2 n + 1} C_{0} = 2^{2 n} .$

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