First slide

Combinations

Question

For $2 \leq r \leq n, (\begin{matrix} n \\ r \end{matrix}) + 2 (\begin{matrix} n \\ r - 1 \end{matrix}) + (\begin{matrix} n \\ r - 2 \end{matrix})$ is equal to:

Easy

A

$(\begin{matrix} n + 1 \\ r - 1 \end{matrix})$
B

$2 (\begin{matrix} n + 1 \\ r - 1 \end{matrix})$
C

$2 (\begin{matrix} n + 2 \\ r \end{matrix})$
D

$(\begin{matrix} n + 2 \\ r \end{matrix})$

Solution

Expression $= C (n, r) + 2. C (n, r - 1) + C (n, r - 2)$
                      $= [C (n, r) + C (n, r - 1)] + [C (n, r - 1) + C (n, r - 2)]$
                      $= C (n + 1, r) + C (n + 1, r - 1)$
                      $= C (n + 2, r)$

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