combination with specifications

[misstokyo]

Hello, everyone. I was hoping you guys could help me out. I'm about to take this huge college entrance exam in two months and I bought this reviewer at a bookstore - the problem, however, is that the reviewer I bought only posts the ANSWERS to the questions without explanations.

How do you solve combination problems when they have specifications?

For example -
In how many ways can a comittee of five be chosen out of 15 people, 7 of which are males and 8 of which are females, if one male and one female are to be included in each selection?
(according to the answer key, the answer is 16,106)

or..

A shipment of 14 personal computers contains 4 defective units. In how many ways can a company purchase 5 of these units and receive at least 2 of the defective units?
(according to the answer key, the answer is 910)

any help would be great.

[maggiemagnet]

In how many ways can a comittee of five be chosen out of 15 people, 7 of which are males and 8 of which are females, if one male and one female are to be included in each selection?

In how many ways can you choose the one required male? In how many ways can you choose the one required female? In how many ways can you choose the remaining three members? Multiplying, what value do you get?

A shipment of 14 personal computers contains 4 defective units. In how many ways can a company purchase 5 of these units and receive at least 2 of the defective units?

To receive "at least two", you will receive two or three or four. The opposite possibility is receiving zero or one. So one method would be:

In how many ways can you chose none of the defective units from the four available? In how many ways can you choose the remaining five units from the remaining ten?

In how many ways can you choose one of the defective units from the four available? In how many ways can you choose the remaining four units from the remaining ten?

Subtract these values from the total number of ways to choose five of fourteen.

[misstokyo]

In how many ways can a comittee of five be chosen out of 15 people, 7 of which are males and 8 of which are females, if one male and one female are to be included in each selection?

In how many ways can you choose the one required male? In how many ways can you choose the one required female? In how many ways can you choose the remaining three members? Multiplying, what value do you get?

there are 7 choices for the one required male, 8 choices for the one required female, and which leaves 13 people to choose from for the third spot, 12 people to choose from for the fourth, and 11 for the last spot. Multiplying 7 x 8 x 13 x 12 x 11, I got 96,096 - but the answer key in my book says the answer should be 16,106.

Help!

[maggiemagnet]

On what basis are you finding the number of ways that they could be picked in order? Isn't this a combination, so the order doesn't matter?

[misstokyo]

On what basis are you finding the number of ways that they could be picked in order? Isn't this a combination, so the order doesn't matter?

oh. right. I forgot about that. *embarrased* well anyway, would it be okay to ask for a detailed explanation as to how to solve these kinds of combination problems? I've been looking around the net and all I've seen are the "normal" or "regular" combinations taken r at a time - I haven't seen any website mentioning anything about specifications.