Permutations and combinations

When we talk of permutations and combinations in everyday talk we often use the two terms interchangeably. In mathematics, however, the two each have very specific meanings, and this distinction often causes problems.
In brief, the permutation of a number of objects is the number of different ways they can be ordered; i.e. which is first, second, third, etc. If you wish to choose some objects from a larger number of objects, the way you position the chosen objects is also important. With combinations, on the other hand, one does not consider the order in which objects were chosen or placed, just which objects were chosen. We could summarise permutations and combinations (very simplistically) as

Permutations - position important (although choice may also be important)

Combinations - chosen important,
which may help you to remember which is which.

As mentioned above, there is an important difference between permutations and combinations. In this case, for permutations the order of events is important: order 1 is different from order 2. For combinations, however, it does not matter which picture was hung first. In this example there are two permutations (A, B ≠ B, A), but only one combination (A, B = B, A).
Another way that you may find useful to help you remember is to consider a combination lock. On combination locks you have to turn dials with numbers on so a particular number is given, e.g. '1, 2, 3, 4'. But they do not unlock when if the order is changed (e.g. 2, 1, 3, 4). In this case the order is important. So combination locks should not be called combination locks but 'permutation' locks.

In the same way that permutations have shorthand, combinations have similar shorthand. All we have to do is divide the number of permutations by the number of permutation in each set. So, the right-hand side of the following equation is the same as the equation for the number of permutations except for an additional r! term in the divisor (which corrects for the number of permutations of each set). Note, also, that the P (for permutation) is replaced by C (for combination).

If you have a scientific calculator you should see these labelled (on some calculators they are separate keys, on others they are second-function keys). In general, you enter the number of items to choose from (n) then the nCr or nPr button and then the number of items to choose (r).

1 comment:

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