Permutations: Alternative Proofs of Elementary Formulas

Extract

The number of the r-permutations of n things is the same as the number of ways of adding r things to a row which originally contains n - r other things.

For suppose the n letters A, B, C … N to be placed in a row, then each permutation consisting of r letters may be indicated by placing below those r letters in the row, the digits 1, 2, 3 … r, to indicate the order they have in the permutation. We may suppose zeros placed below all the remaining n - r letters. Thus it is clear that the number of the r-permutations of n things is the same as the number of ways in which r numbers can be added to a row originally containing n - r zeros.