What is a permutation

In a previous post I wrote about what a factorial is. Its the number of ways n distinct positions can be filled. Or, the number of ways n distinct elements can be arranged in n distinct positions.

Now the permutation. The number of elements is more than the number of positions to be filled or vice versa. If given 6 distinct elements, we need to find out how many ways we can fill up 4 distinct positions, that is a permutation.

The problem is similar to the factorial but in this case we only care about 4 of the 6 elements. The factorial solution only works when the number of positions to be filled coincides with the number of elements to fill up the positions. Otherwise, we need a different formula.

So we find the number of ways to distinctly fill 6 positions. Then from that we take off the number of ways to fill 2 positions. Since filling 6 evaluates to 6x5x4x3x2x1 but we only care about the first four positions. The trailing (2x1) is cut off by dividing 6! by (6-4)!.

The formula hence becomes:

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

Where n is the total number of slots to fill or the total number of elements and r is what number out of the total we shall be working with.

Some interesting applications of permutations

• Binomial theorem (Pascal's triangle)