Factorial, Permutation, Combination

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The number of ways we can arrange $n$ distinct objects is given by:

$$n!=n\times (n-1)\times (n-2)..\mathrm{.2}\times 1$$

The number of ways we can arrange 5 distinct objects is:

$$5!=5\times 4\times 3\times 2\times 1$$

Syntax for factorial:

import numpy as np
np.math.factorial(n)

The number of ways we can arrange $n$ distinct objects, taking them $r$ at a time is given by:

$$nPr=\frac{n!}{(n-r)!}$$

import numpy as np
np.math.factorial(n)/np.math.factorial(n-r)

The number of distinct combinations of $n$ distinct objects that can be formed by taking $r$ at a time is given by:

$$nCr=\frac{n!}{r!(n-r)!}$$

import numpy as np
np.math.factorial(n)/(np.math.factorial(r)*np.math.factorial(n-r))

Example: For $n=6$ and $r=3$, compute the following:

1. $n!$
2. $nPr$
3. $nCr$

import numpy as np
np.math.factorial(6)

720

Interpretation: 6!=720. There 720 different arrangement of 6 distinct objects.

import numpy as np
np.math.factorial(6)/np.math.factorial(6-3)

120.0

Interpretation: there are 120 permutations of 6 objects taking 3 at a time.

import numpy as np
np.math.factorial(6)/(np.math.factorial(3)*np.math.factorial(6-3))

20.0

Interpretation: There are 20 combinations (without regard to order) of 6 objects taking 3 at a time.

Adapted from Python for Introductory Statistics, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.