Combinations

Combinations are selections. From a group of 10 objects, five are selected at random. The group of five objects is called a combination. Combinations are different from permutations in that the order in the selected objects are arranged matters in permtuations but not in combinations. A combination of objects can be arranged in different orders giving rise to permutations. If there are n objects, and r are selected from them at random, then each combination of r objects can be arranged in different orders to form r! permutations.

We know that the formula for permutations for n objects taken r at a time is

P(n, r) = n! / (n - r)!.

Since for each combination of r objects selected at random from a group of n objects, the number of permutations is r!, therefore the total number of permutations is r! times the total number of combinations.

Now let the total number of combinations of n objects taken r at a time be x. By the above derivation,

x * r! = P(n, r)

That is,

x * r! = n! / (n - r)!

Therefore,

x = n! / r!(n - r)!

Therefore the total number of combinations of n objects taken r at a time is

C(n, r) = n! / r!(n - r)!