Question

A certain mathematics contest has a peculiar way of giving prizes. Five people are named as Grand Prize winners, but their finishing order is not listed. Then from among the other entrants, a 6th place, 7th place, 8th place, 9th place, and 10th place winner are each named. If 22 people enter this year, how many complete award announcements are possible?

Answer 1

Answer:

19554575040

Explanation:

The first five positions to be named as Grand Prize winners are not give any order.

Therefore, the number of ways they can be selected is

$_{5}^{22}\textrm{C}$

$\frac{22!}{17!5!}$

Now the remaining 5 positions are selected from the remaining 17 entries. So they can be ordered as,

$_{5}^{17}\textrm{P}$

$\frac{17!}{12!}$

17 x 16 x 15 x 14 x 13 ways of ordering

Therefore, the number of ways in which the entries can be announced is

$\frac{22!}{17!5!}\times 17\times 16\times 15\times 14\times 13$

= 19554575040 number of ways

Thus the announcement can be made in 19554575040 number of ways.