First slide

Combinations

Question

If $S = \sum_{r = 0}^{m}^{n + r} C_{k}$ , then

Moderate

A

$S +^{n} C_{k + 1} =^{n + m} C_{k + 1}$
B

$S +^{n} C_{k + 1} =^{n + m + 1} C_{k + 1}$
C

$S +^{n} C_{k} =^{n + m} C_{k}$
D

None of these

Solution

$\begin{matrix} S = \sum_{r = 0}^{m}^{n + r} C_{k} \\ =^{n} C_{k} +^{n + 1} C_{k} +^{n + 2} C_{k} + \dots +^{n + m} C_{k} \\ =^{n} C_{k + 1} +^{n} C_{k} +^{n + 1} C_{k} +^{n + 2} C_{k} + \dots +^{n + m} C_{k} -^{n} C_{k + 1} \\ =^{n + 1} C_{k + 1} +^{n + 1} C_{k} +^{n + 2} C_{k} + \dots +^{n + m} C_{k} -^{n} C_{k + 1} \\ ∴ S =^{n + m + 1} C_{k + 1} -^{n} C_{k + 1} \end{matrix}$

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