N factorial - general question

[michonamona]

Hello everyone!

Homework Statement

n!/(n-r)! = n(n-1)(n-2)...(n-r+1)

where r is the number of objects we want from n distinct objects (3 billiard balls out of 16)

I don't understand what the last term in the expansion means, the (n-r+1). For example, suppose we have 5 distinct objects and we want to choose 2. This means that n = 5 and r = 2

5x4x3x2x1x(5-2+1)

is this correct?

Thank you for your help

 

[cronxeh]

16!/(16-3)! = 3360 which is P(n,r)

If you want C(n,r)=P(n,r)/r!

 

[Mathnerdmo]

The notation n(n-1)(n-2)...(n-r+1) actually means to start at n and multiply all the numbers below it until we reach n-r+1.

Therefore with your example where n=5 and r=2, we have that
n-r+1 = 5-2+1 = 4
and we multiply
5 * 4 = 20
where we don't include any factor below 4.

Similarly, if we had n=5 and r=3, then we'd find
n-r+1 = 5-3+1 = 3
so when we evaluated n(n-1)(n-2)...(n-r+1), we'd have
5 * 4 * 3 = 60
(once again, note that we don't include any factor below n-r+1 = 3)

Hope that clarifies things.

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Additionally, like cronxeh mentioned, that expression counts permutations. In your example of the billiard balls, we'd want to count combinations, unless you're also giving some order to the choice.

 

[michonamona]

Awesome, thanks guys.