Combinations

Combinations are similar to permutations except that for combinations, the order within the combination does not matter. The six permutations below count as a single combination.

All six of the permutations above involve arrangements of blue, red and yellow cubes. As the order doesn’t matter in combinations, these six permutations count as a single combination of blue, red and yellow.

In the situation where we have five coloured cubes and three positions to fill, how many combinations of the cubes are there?

We calculate this by first working out how many permutations there are and then dividing the number of permutations by the number of ways of arranging three coloured cubes.

The number of permutations are and there are 6 ways to arrange three cubes so to calculate the number of combinations we need to divide this number by 6 to get

We calculate the 6 ways of arranging three cubes by the usual method. There are 3 ways to choose the first cube, two ways to choose the second and 1 way to choose the last cube. This is 3 factorial (3!).