1. [Combinations//Permutations] Item Arrangments

Johnathan has 18 different trophies on his shelf. Four of the trophies are for racing, nine are for high jumping, five are for cycling.
How many different displays can be made if:
1, Six trophies are chosen?
$\displaystyle nPr(18,6)=13366080$
2, Six trophies are chosen, two from each activity?

2. Originally Posted by Cthul
How many different displays can be made if:
1, Six trophies are chosen?
$\displaystyle nPr(18,6)=13366080$
2, Six trophies are chosen, two from each activity?

3 pairs of trophies are chosen.
1 pair must be racing trophies, so "select" 2 from 4 to find out how many pairs of racing trophies you can have.

Another pair must be high-jump trophies, so "select" 2 from 9.

Another pair must be cycling trophies, so "select" 2 from 5.

You can combine any pair of racing trophies with any pair of high-jump trophies,
hence multiply the selections.
Those 4 can go with any pair of cycling trophies, so multiply again.

Use nCr for the selections.

For your Q1, you've calculated displays with the 6 trophies in different positions.
Is this feasible?

3. Er..
I can do Question 1, I just thought I might add it in to let readers know that I know a some parts.

So it's..
$\displaystyle nCr(5,2)\times nCr(9,2)\times nCr(4,2)\times6!$... I got it right. Thanks.