Combinations Question

[fishbiscuit]

Question:
A committee of 7 politicians is chosen from 10 liberal members, 8 labor members and 5 independents. In how many ways can this be done so as to include exactly 1 independent and at least 3 liberal members and at least 1 labor member?

Answer: 73080

I'm just confused - I can't even explain the problems I'm having with this question. Help will be greatly appreciated!

(question is from J.B. Fitzpatrick 3 unit light blue book)

[jamonwindeyer]

Hello! The trick here is to break this decision into your three pieces:

- Pick the independent member
- Pick your labour members
- Pick your liberal members

Now, we need to figure out how many different ways we can do each of these things and keep with the conditions. There are three:

- Pick 1 independent, 3 liberals, and 3 labor
- Pick 1 independent, 4 liberals, and 2 labor
- Pick 1 independent, 5 liberals, and 1 labor

These are the three things we need to consider. Each one we can represent with a combination! The first one, for example, would be:



The first combination is picking 1 independent out of five, the next is picking three liberals out of 10, and the last is picking three labor out of 8.

You do one of these for each of the three cases above (so three sets of three combinations), and add them up!! Does this help?

[fishbiscuit]

Yes, this does help thank you very much!

[jamonwindeyer]

Yes, this does help thank you very much!

Excellent, very welcome!!