Permutations
When the elements of a set are arranged in a definite order, the arrangement is called a permutation of the elements. The number of permutations of objects is
The number of possible orderings of objects taken from a set of is given by:
That is, count backwards starting from , writing down the numbers as you count, until you've written down numbers. Then multiply them all together.
Example:
Suppose you're a television programmer, and you have five half-hour shows to choose from, but only three time slots. How many different programs are possible?
Using the permutations formula, we have:
To see why this works, name the shows A, B, C, D, and E, and make a list:
ABC ABD ABE ACB ACD ACE ADB ADC ADE AEB AEC AED |
BAC BAD BAE BCA BCD BCE BDA BDC BDE BAC BAD BAE |
CAB CAD CAE CBA CBD CBE CDA CDB CDE CEA CEB CED |
DAB DAC DAE DBA DBC DBE DCA DCB DCE DEA DEB DEC |
EAB EAC EAD EBA EBC EBD ECA ECB ECD EDA EDB EDC |
In this case, there are choices for the first program, choices for the second program, and choices for the last program. So the answer is:
If there were programs and time slots, we would have: