1. ## combination

$8$ bottles of zinfandel

$10$ bottles of merlot

$12$ bottles of cabaret.

(a) If he wants to serve $3$ bottles of zinfandel and serving order is important, how many ways are there to do this?

$\frac{8!}{5!} = 336$

(b) If $6$ bottles of wine are to be randomly selected from the $30$, how many ways are there to do this?

$\binom{30}{6} = 593,775$

(c) If $6$ bottles are randomly chosen, how many ways are there two bottles of each variety being chosen?

(d) If $6$ bottles are randomly chosen, what is the probability that this results in two bottles of each variety being chosen.

(e) If $6$ bottles are randomly chosen, what is the probability that they are all the same variety?

2. Originally Posted by shilz222

(c) If $6$ bottles are randomly chosen, how many ways are there two bottles of each variety being chosen?
${ 8 \choose 2} {10 \choose 2} {12 \choose 2}$

(d) If $6$ bottles are randomly chosen, what is the probability that this results in two bottles of each variety being chosen.
$\frac {{ 8 \choose 2} {10 \choose 2} {12 \choose 2}}{{ 30 \choose 6}}$

(e) If $6$ bottles are randomly chosen, what is the probability that they are all the same variety?
$\frac {{ 8 \choose 6} + {10 \choose 6} + {12 \choose 6}}{{30 \choose 6}}$

why is that?

3. There are $\binom{8}{6}$ ways to choose all zinfandel, etc..