What is Permutations
Consider the simple problem of placing three balls colored red, blue, and white in 10 boxes numbered 1, 2, 3….. 10. We want to know the number of distinct ways in which the balls can be placed in the boxes, if each box can hold only one ball. Let us place the ball one at a time, beginning with the red ball, then the blue boxes, the blue ball can be placed in any of the eight remaining boxes, the total number of distinct ways to place these balls is 10 × 9 × 8 = 720.
The result of this numerical example can be generalized immediately: Suppose we are to place r distinctly colored balls in n distinctly numbered boxes with the condition that a box only one ball. Since the first ball can be placed in any one of the n boxes, the second ball can be placed in any one of the remaining (n – 1) boxes, …., and the rth ball can be placed in any one of the remaining (n –r + 1) boxes, the total number of distinct ways to place the balls is
n(n – 1) (n -2) …. (n – r + 1)
which can also be written as:
n!/(n – r)!
we use the notation P(n, r) for the quantity n(n – 1) (n – 2) …. (n – r + 1).
The following examples show that the problem of placing balls in boxes is not as uninteresting as it might seem.
Example: In how many ways can three examinations be scheduled within a five-day period so that no two examinations are scheduled on the same day? Considering the three examinations as distinctly colored balls and the five days as distinctly numbered boxes, we obtain the result 5 × 4 × 3 = 60.
Comments
Post a Comment