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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 n objects is n !

The number of possible orderings of m objects taken from a set of n is given by:

P n m = n × ( n 1 ) × ( n 2 ) × × ( n m + 1 ) = n ! ( n m ) !

That is, count backwards starting from n , writing down the numbers as you count, until you've written down m 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:

P 5 3 = 5 × 4 × 3 = 60

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 5 choices for the first program, 4 choices for the second program, and 3 choices for the last program. So the answer is:

P 5 3 = 5 × 4 × 3 = 60

If there were 8 programs and 4 time slots, we would have:

P 8 4 = 8 × 7 × 6 × 5 = 1680

 

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