First slide

Combinations

Question

$^{m} C_{r + 1} + = \sum_{k = m}^{n}^{k} C_{r} =$

Moderate

A

$^{n} C_{r + 1}$
B

$^{n + 1} C_{r + 1}$
C

$^{n} C_{r}$
D

None of these

Solution

$^{m} C_{r + 1} + \sum_{k = m}^{n}^{k} C_{r}$
$\begin{array}{l} =^{m} C_{r + 1} +^{m} C_{r} +^{m + 1} C_{r} + \dots +^{n - 1} C_{r} +^{n} C_{r} \\ =^{m + 1} C_{r + 1} +^{m + 1} C_{r} + \dots +^{n - 1} C_{r} +^{n} C_{r} \\ =^{m + 2} C_{r + 1} +^{m + 2} C_{r} + \dots +^{n - 1} C_{r} +^{n} C_{r} \end{array}$
$⋮$ $⋮$ $⋮$ $⋮$ $⋮$
$=^{n} C_{r + 1} +^{n} C_{r} =^{n + 1} C_{r + 1}$

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