A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 9 of cabernet, all from different wineries. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?

For this problem, we use the formula for combination. When the notation is written as nCr, that is equivalent to the equation below:

nCr = n!/r! (n-r) !

where

n is the total number of objects

r is the number of one type of object

Using the fundamental counting principle, we multiple each nCr equation for each type of bottle. The solution is:

Number of ways = 8C2*10C2*9C2 = 45,360 ways

This problem can be simply solved by calculating for the product of each combination of each wine. There are 8C2 ways to pick from 8 bottles of zinfandel, 10C2 ways to pick from merlot and 10C9 ways to pick from cabernet.

Total number of ways = 8C2 * 10C2 * 9C2

Total number of ways = 45,360