First slide

Combinations

Question

If $^{n + 1} C_{r + 1} :^{n} C_{r} :^{n - 1} C_{r - 1} = 11 : 6 : 3$ then the value of $n$ is

Moderate

A

12
B

10
C

18
D

21

Solution

As $(^{n + 1} C_{r + 1}) = \frac{(n + 1)}{r + 1} (^{n} C_{r}) \frac{(^{n + 1} C_{r + 1})}{(^{n} C_{r})} = \frac{(n + 1)}{r + 1} = \frac{11}{6} \Rightarrow 6 n + 6 = 11 r + 11 \Rightarrow 6 n = 11 r + 5 \frac{(^{n} C_{r})}{(^{n - 1} C_{r - 1})} = \frac{n}{r} = \frac{6}{3} \Rightarrow n = 2 r 6 n = 11 r + 5 \Rightarrow 12 r = 11 r + 5 \Rightarrow r = 5 n = 10$

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