Combination

Probability Level 3

$^{ n }{ { { C } } }_{ x }:^{ n }{ { { C } } }_{ x+1 }:^{ n }{ { { C } } }_{ x+2 }=2:3:4$

Find $n \times x$

The answer is 442.

1 solution

Chew-Seong Cheong
Dec 26, 2014

$^nC_x:^nC_{x+1}:^nC_{x+2} = 2:3:4$

$\Rightarrow \dfrac {n!}{x!(n-x)!} : \dfrac {n!}{(x+1)!(n-x-1)!} : \dfrac {n!}{(x+2)!(n-x-2)!} = 2:3:4$

$\Rightarrow \begin {cases} \dfrac {(x+1)!(n-x-1)!} {x!(n-x)!} = \dfrac {x+1} {n-x} = \dfrac {2}{3} & \Rightarrow 5x = 2n - 3 & ...(1)\\ \dfrac {(x+2)!(n-x-2)!} {(x+1)!(n-x-1)!} = \dfrac {x+2} {n-x-1} = \dfrac {3}{4} & \Rightarrow 7x = 3n - 11 & ...(2) \end {cases}$

$3 \times$ eqn 1 - $2\times$ eqn 2:

$\Rightarrow \begin {cases} 15x = 6n - 9 & ...(1a)\\ 14x = 6n - 22 & ...(2a) \end {cases} \Rightarrow x = 13 \quad \Rightarrow n = 34$

Therefore the required answer: $n\dot{} x = 34\times 13 = \boxed {442}$

Abdulrahman, it was unclear where $n . x$ meant "place a decimal point there" or "multiply these numbers". Based on the above solution, I have edited it to $n \times x$ .

Calvin Lin Staff - 6 years, 5 months ago

Combination

The answer is 442.

1 solution

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