Statistics - Permutation

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, suppose we have a set of three letters: A, B, and C. we might ask how many ways we can arrange 2 letters from that set.

Permutation is defined and given by the following function:


${^nP_r = \frac{n!}{(n-r)!} }$

Where −

  • ${n}$ = of the set from which elements are permuted.

  • ${r}$ = size of each permutation.

  • ${n,r}$ are non negative integers.


Problem Statement:

A computer scientist is trying to discover the keyword for a financial account. If the keyword consists only of 10 lower case characters (e.g., 10 characters from among the set: a, b, c... w, x, y, z) and no character can be repeated, how many different unique arrangements of characters exist?


Step 1: Determine whether the question pertains to permutations or combinations. Since changing the order of the potential keywords (e.g., ajk vs. kja) would create a new possibility, this is a permutations problem.

Step 2: Determine n and r

n = 26 since the computer scientist is choosing from 26 possibilities (e.g., a, b, c... x, y, z).

r = 10 since the computer scientist is choosing 10 characters.

Step 2: Apply the formula

${^{26}P_{10} = \frac{26!}{(26-10)!} \\[7pt] \ = \frac{26!}{16!} \\[7pt] \ = \frac{26(25)(24)...(11)(10)(9)...(1)}{(16)(15)...(1)} \\[7pt] \ = 26(25)(24)...(17) \\[7pt] \ = 19275223968000 }$