n P r ^nP_r n C r ^nC_r

Probability Level 2

n P r = 4896 n C r = 816 \LARGE {^nP_r=4896\\^nC_r=816}

What's the value of ( n − r ) (n-r) ??


The answer is 15.

Md Mehedi Hasan by Md Mehedi Hasan , 22, Bangladesh

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2 solutions

Md Mehedi Hasan
Nov 25, 2017

Here, n C r = 816 ^nC_r=816

And n P r = r ! × n C r = 4896 ⇒ r ! = 4896 816 = 6 = 3 ! ∴ r = 3 ^nP_r=r!\times ^nC_r=4896\\\Rightarrow r!=\frac{4896}{816}=6=3!\\\therefore r=3

Again n P r = n ! ( n − r ) ! = n ! ( n − 3 ) ! = 4896 ⇒ n ( n − 1 ) ( n − 2 ) = 4896 ⇒ n 3 − 3 n 2 + 2 n − 4896 = 0 ^nP_r=\frac{n!}{(n-r)!}=\frac{n!}{(n-3)!}=4896\\\Rightarrow n(n-1)(n-2)=4896\\\Rightarrow n^3-3n^2+2n-4896=0

Solving that we get n = 18 n=18

So, the answer is ( n − r ) = 18 − 3 = 15 (n-r)=18-3=\boxed{15}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

No need to solve that cubic equation... just factorise 4896... we get 18×17×16...

Hence 18 is the value of n

Skanda Prasad - 3 years, 6 months ago

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But how could you know 4896 = 18 ⋅ 17 ⋅ 18 4896=18\cdot 17\cdot 18 at first?

Md Mehedi Hasan - 3 years, 6 months ago

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Easy...I'll post it as a solution...you can see it there...

Skanda Prasad - 3 years, 6 months ago
Skanda Prasad
Nov 26, 2017

@Md Mehedi Hasan , looking at 4896, we must get to now that it's divisible by 24... after that things are easy...

Or even 48 is fine, but to my head, 24 struck before 48...so I used 24...nothing wrong with 48...

1 Helpful 0 Interesting 0 Brilliant 0 Confused

well, here you have to find the divisor. It's also a good process. But you can solve cubic equation with calculator.

Md Mehedi Hasan - 3 years, 6 months ago

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Bro... Always try to stay away from calculator unless and until it's absolutely necessary... For engineering level use of calc is fine but for problems of these kind, calc is not at all necessary. Mental calculations are better... and more effective than the use of a calculator.

And it's nothing like a treasure hunt to find the divisor... where so many obstacles are there, well all I mean to say is that it's an easier and better way to do this than using a calc

Skanda Prasad - 3 years, 6 months ago

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Yeah, you are correct. Go ahead bro...

Md Mehedi Hasan - 3 years, 6 months ago

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