Additional Mathematics - Permutations and Combinations (formulae)

Factorial n
n! = n x (n-1) x (n-2) x (n-3) x ...
Eg, 8!
= 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
= 40,320

Eg, How many ways to arrange 6 books on a shelf:
Ans : 6! ways

Permutations
Arrangement of a list of objects where order does matter.

nPr = n!/[(n-r)!]
Eg 8P3
= 8!/[(8 - 3)!]
= (8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ( 5 x 4 x 3 x 2 x 1)
= 8 x 7 x 6
= 336

Eg How many ways to pick a President, Vice President and Treasurer from a group of 10? The order does matters on who is the President, Vice President or Treasurer.
Ans 10P3 = 720
The permutation formulae denominator removes 4th to 10th order with no title.

Eg List down your 3 favorite desserts, in order, from a menu of 10.
Ans: 10P3 = 720.
The permutation formulae denominator removes the 4th to 10th desert which are not your favourite.

Combinations
Arrangement of a group of objects where order does not matter.

nCr = n!/[(n-r)! x r!]
Eg 8C3
= 8!/[(8-3)! x 3!)
= 8!/(5! x 3!)
= 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 /[(5 x 4 x 3 x 2 x 1) x (3 x 2 x 1]

Eg Picking a team of 3 people from a group of 10. The order is not important.
10C3 = 120

Eg Choose 3 desserts from a menu of 10. The order of which is your favourite is not important.
10C3 = 120.

Although you can use a scientific calculator to derive at the above answers. It is important that you understand the difference in the formulaes of permutation and combination in order to apply it.
= 56