Chapter 8.5, Problem 43E

To determine

To calculate: total number of different orders for top three designs from total received entries.

Answer to Problem 43E

Number of different orders for top three designs is 456456.

Explanation of Solution

Given information:

1. Total number of entries received in a contest = 78
2. Awards are given for first, second and third positions.

Formula used:

  $P{}_r=\frac{n!}{(n-r)!}$

Where $n$ = Total number of objects and $r$ = number of objects selected at a time.

Calculation:

Need to calculate number of different orders of top three designs from 78 designs. As order of first, second and third position is required given scenario is case of permutation.

Total number of entries received in contest that is $n=78$

Award are given for first, second and third position that is $r=3$

Substitute $n=78$ and $r=3$ in formula $P{}_r=\frac{n!}{(n-r)!}$

  $P{}_3=\frac{78!}{(78-3)!}$

  $=\frac{78\times 77\times 76\times 75!}{75!}$

  $=78\times 77\times 76$

  $=456456$

Hence number of different orders for top three designs is 456456.