N factorial - general question

In summary, the formula n!/(n-r)! can be used to find the number of ways to choose r objects from n distinct objects. The notation n(n-1)(n-2)...(n-r+1) means to start at n and multiply all the numbers below it until we reach n-r+1. This formula counts permutations, but for combinations, we need to divide by r!.
  • #1
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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
 
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  • #2
16!/(16-3)! = 3360 which is P(n,r)

If you want C(n,r)=P(n,r)/r!
 
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  • #3
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.

-------

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.
 
  • #4
Awesome, thanks guys.
 

What is the concept of N factorial?

N factorial, denoted as N!, is the product of all positive integers from 1 to N. It is a mathematical concept commonly used to calculate the number of ways to arrange a set of objects or to find the total number of outcomes in a probability experiment.

How is N factorial calculated?

N factorial can be calculated by multiplying all the positive integers from 1 to N. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. For larger values of N, it is often easier to use a calculator or a computer program to calculate the factorial.

What are the applications of N factorial?

N factorial has many applications in mathematics, including combinatorics, probability, and statistics. It is also used in fields such as computer science, physics, and economics to solve various problems and equations.

Is there a limit to how large N can be in N factorial?

Technically, there is no limit to how large N can be in N factorial. However, for practical purposes, the value of N should be limited to numbers that can be accurately represented and calculated by a computer. For example, 20! is approximately 2.4 x 10^18, which is close to the maximum value that can be represented by a 64-bit computer.

What is the relationship between N factorial and the gamma function?

The gamma function, denoted as Γ(n), is a generalization of the concept of N factorial for non-integer values of n. The gamma function is defined as the integral from 0 to infinity of x^(n-1)e^-x dx. When n is a positive integer, Γ(n) = (n-1)!. Therefore, N factorial can be thought of as a special case of the gamma function for positive integer values of n.

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