An important application of the fundamental counting principle is to find the number of ways a set of elements can be ordered. List the number of ways the three letters a , b , and c can be arranged without duplicating the different orders. The six arrangements are abc , acb , bca , bac , cab , and cba . Any time a definite order of objects is required permutations may be used to determine the possible arrangements. A permutation is an arrangement of things in a definite order.
P r represents a permutation of n elements taken r at a time.
Another way of finding 8 P 3 is to multiply the three consecutive numbers from 8 down to 6. The 8 preceding the P tells how many items or elements are involved, the 3 following the P tells how many consecutive numbers to multiply. The tree diagram on page 415 depicts the number of ways five horses could finish a race in five different positions. Drawing a tree diagram of this magnitude is very time consuming, therefore the use a permutation seems more advantageous. There are 15 choices for the gold medal winner.