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

Step-by-step explanation:

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

$P_{n,r} = \frac{n!}{(n-r)!}$

$P_{9,3} = \frac{9!}{(9 - 3)!}$

$P_{9,3} = \frac{9.8.7.6!}{6!}$

$P_{9,3}$ = 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:

$C_{n,r} = \frac{n!}{r!(n - r)!}$

$C_{30,6} = \frac{30!}{6!.24!}$

$C_{30,6} = \frac{30.29.28.27.26.25.24!}{6.5.4.3.2.1.24!}$

$C_{30,6}$ = 593775

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