1. ## Probability

We have 8 bottles of Zinfandel. 10 of Merlot, and 12 of Cabernet.

If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?
$P=8!/(8-3)! = 336$

If 6 bottles of wine are to be randomly selected for the 30 for serving, how many ways are there to do this?
$P=30!/6!(30-6)! = 593,775$

If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?
$P = {{\binom{8}{2}} {\binom{12}{2}} {\binom{10}{2}}} = 83,160
$

If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen?
$P = {{\binom{8}{2}} {\binom{12}{2}} {\binom{10}{2}}}/{\binom{30}{6}} = .1400
$

If 6 bottles are randomly selected, what is the probability that all of them are the same variety.
Not even sure how to start this one. Any help?

Are these right?

2. The last one:
$\displaystyle \frac{\binom{8}{6}+\binom{10}{6}+\binom{12}{16}}{\ binom{30}{6}}$

3. Originally Posted by Plato
The last one:
$\displaystyle \frac{\binom{8}{6}+\binom{10}{6}+\binom{12}{16}}{\ binom{30}{6}}$
How did you get 6, 6, and 16?

Also, I have another general question. When should I use the following formulas:
$P = n!/(n-k)!$
and
$P = n!/(n-k)!*k!$

Neither my book/teacher really explain the difference between these two. Is the first just the number of combinations, and the second a percentage of possibilities?

thanks, btw is #4 really correct? I think I get a different number than the answer, much appreciated!

4. Originally Posted by Plato
The last one:
$\displaystyle \frac{\binom{8}{6}+\binom{10}{6}+\binom{12}{16}}{\ binom{30}{6}}$
It should have been $\displaystyle \frac{\binom{8}{6}+\binom{10}{6}+\binom{12}{6}}{\b inom{30}{6}}$
It was a typo.

5. Originally Posted by Plato
It should have been $\displaystyle \frac{\binom{8}{6}+\binom{10}{6}+\binom{12}{6}}{\b inom{30}{6}}$
It was a typo.
thank you, also does 4 seem correct?

6. Originally Posted by hansel13
thank you, also does 4 seem correct?
YES

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# if 6 bottles of wine are to be randomly selected from the 29 for serving, how many ways are there to do this?

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