First slide

Combinations

Question

If $^{n} C_{r} :^{n} C_{r + 1} :^{n} C_{r + 2} = 1 : 2 : 3$ then $r$ is equal to

Moderate

A

5
B

4
C

3
D

0

Solution

$\frac{2}{1} = \frac{^{n} C_{r + 1}}{^{n} C_{r}} = \frac{n - r}{r + 1}$
and $\frac{3}{2} = \frac{n - r - 1}{r + 2}$
$\begin{array}{l} \Rightarrow \frac{3}{2} = \frac{2 (r + 1) - 1}{r + 2} [∵ n - r = 2 (r + 1)] \\ \Rightarrow 3 r + 6 = 4 r + 2 \Rightarrow r = 4 . \end{array}$

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