First slide

Combinations

Question

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

Moderate

A

2²ⁿ
B

2²ⁿ⁺¹
C

2²ⁿ – 1
D

None of these

Solution

If we choose k (0 ≤ k ≤ 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
$\begin{array}{l} = \sum_{k = 0}^{n}^{2 n + 1} C_{n - k} =^{2 n + 1} C_{n} +^{2 n + 1} C_{n - 1} + \dots +^{2 n + 1} C_{0} \\ = 2^{2 n} \end{array}$

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