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Permutations and Combinations

Factorial

In probability, a system of counting occurs frequently which involves a series of multiplications of the form:

$$n! = n ⋅ (n - 1) ⋅ (n - 2) ⋅ (n - 3) .... 2 ⋅ 1$$

This function has the name factorial, and has the symbol '!'.

For example, 6! = 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 ( = 720)

1! = 1, and 0! = 1 (careful: 0! is NOT equal to zero)

Factorials can be divided by other factorials: ${6!}/{4!} = {6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1}/{4 ⋅ 3 ⋅ 2 ⋅ 1} = {6 ⋅ 5} = 30$

Permutations

A permutation of a group of symbols is this group in any arrangement in a definite order.

e.g the numbers 1, 2, 3 may be organised in the orders: 123, 132, 213, 231, 312, 321.

There are 6 (3!) possible permutations for 3 symbols. There are 24 (4!) possible permutations for 4 symbols, and 120 possible permutations for 5 (5!) symbols.

Rule: n symbols may be arranged n! ways.

Permutations of sub-sets

The number of ways 5 items may be arranged, taking 3 at a time is: ${5!}/{2!} = 5 ⋅ 4 = 20$

Rule: n symbols taken r at a time may be arranged in ${n!}/{(n - r)!}$ ways.

6 cards taken two at a time form ${n!}/{(n - r)!} = {6!}/{(6 - 2)!} = {6!}/{4!} = 6 ⋅ 5 = 30$ pairs, where the order is important: (1,2) is different to (2,1).

This can be written as:

$$_nP_r = {n!}/{(n - r)!}$$

Combinations

In $_nP_r$ the order of the sub-set of r items is important. In many cases, the order is not relevant. For example, in dealing hands of cards, the player can rearrange the cards he is dealt.

Therefore, if the order of the sub-set is not important, there are fewer combinations possible. This quantity is given the symbol C.

$$_nC_r = {n!}/{{(n - r)!}r!}$$

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