Combinations

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What is a combination?

A combination is a selection of all or part of a set of n objects without repetition, and with regardless to the order in which objects are selected. That is, the order is not important.

For example, suppose you are asked to select a set of 4 different letters from the 26 English alphabet {A, B, C, D,. . . , Z} without regard to the order. Let's say you got

$\left \{ A, B, D, C \right \}, \left \{ C, B, D, A \right \}, \left \{ D, E, H, K \right \}$

Actually, the set {A, B, D, C}, {C, B, D, A} have the same set of letters, even though the 4 letters are selected in different order. Hence, you have selected only two different sets of 4 letters (not 3 sets!).

Notations for combination

The number of combinations in which r objects can be selected from a set of n objects, where repetitions are not allowed and the ordering is not important, is denoted by $C(n, r) \or\ C_{r}^{n}$
$C(n,\:r)=\frac{n!}{r^{!}(n-r)^{!}}$
Notes:
	1.	$\begin{align} C(n&, r) P(n, r)\\ &=r!\\ \end{align}$
	2.
	3.
	4.

Example 6

How many different ways of 4 letters can be selected from the 26 alphabet when repetitions are not allowed and the ordering of letters is not important?

Solution

There are P(26, 4) ways of arranging any 4 letters chosen from the 26 alphabet without repetitions. We can apply the permutation to get

$P(26,\:4)=\frac{26!}{(26-4)!}=\frac{26!}{22!}=358\:800\:ways$