Question

A friend of mine is giving a dinner party. His current wine supply includes 9 bottles of zinfandel, 7 of merlot, and 11 of cabernet (he only drinks red wine), all from different wineries.

(a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?(b) If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?

Answer 1

Answer: a) 783 ways; b) 593775 ways

Explanation:

a) Your friend wants to select 3 zinfandel out of 9 from the supply in a particular order, which means:

`P_(n,r) = (n!)/((n-r)!)`

`P_(9,3) = (9!)/((9 - 3)!)`

`P_(9,3) = (9.8.7.6!)/(6!)`

`P_(9,3)` = 783

In the dinner party, the friend will have 783 ways of serving the zinfandel.

b) Now, your friend want to select 6 bottles out of 30 in no particular order and randomly selected. So:

`C_(n,r) = (n!)/(r!(n - r)!)`

`C_(30,6) = (30!)/(6!.24!)`

`C_(30,6) = (30.29.28.27.26.25.24!)/(6.5.4.3.2.1.24!)`

`C_(30,6)` = 593775

If you select 6 bottles randomly from 30, you will have 593775 ways of doing it.