How do you do this problem?

zombie

A legislative committee consists of 5 Democrats and 4 Republicans. A delegation of 3 is to be selected to visit a small Pacific island republic. How many delegations would include at least 1 Republican?

Thanks.

kevin

There are tow ways of going about doing this problem. The first is to use nCr. This is used when you want to find out the total number of combinations that can be made out of a sample given how large the end group needs to be, and the order does not matter. If the order does matter then we would use nPr, but it doesn't so we wont In other words, we are selecting "r" number of things from a sample consisting of "n" things. The nCr equation goes like this:

n!

________

r! (n-r)!

(you can also just bring up the nCr equation on a calculator and it will do it for you, but im going to show you how the equation works as well)

so, lets say the question was just how many total combinations can be made with a delegation size of 3 out of a committee of 9. we would do this:

9C3

9!

__________

3! (9-3)!

written out it looks like this:

9*8*7*6*5*4*3*2*1

________________

3*2*1 * 6*5*4*3*2*1

reduced it looks like this:

9*8*7

______

3*2

reduced again:

504

____

6

which comes out to 84 possibilities

your question is basically the same, it jest requires one more step. you want to select a group of either 1 rep. and 2 dem. OR 2 rep. and 1 dem. OR 3 rep. Here's how you do that:

the firs possibility can be 1 rep. and 2 dem. so we want to choose 1 rep. out of 4 rep. and 2 dem. out of 5 dem. sooooo:

4C1 * 5C2

if you follow the same equation as above it comes out to

4*10

which equals 40 (remember that)

OR

we could also have 2 rep. out of 4 rep. and 1 dem. out of 5 dem.:

4C2 * 5C1

which comes out to

6*5

which equals 30(remember that as well)

and lastly we could have 3 rep. out of 4 rep.

4C3

which comes out to 4

lastly we need to add all those numbers we remembered up, as all of them contain at least 1 rep.

so 40 + 30 + 4 =

74, that's your answer.

the second way to do it would be to assign each delegate a number (e.g., dem.1, dem.2, rep.5...) and write out the entire sample space, for example:

dem.1, dem.2, rep.1

dem.1, dem.2, rep.2

dem.1, dem.2, rep.3...

and so on, but that would take waaaayy too long and i assure you, the equation is much easier that you may think, just ask your teacher for examples and do them with him and you'll get it fast!

Hope this helped!!!!!

yasmine-reid_60bc3a2a8ca92

C(4,3) will have all republicans

C(4,2) * C(5,1) will have two republicans and 1 democrat

C(4,1) * C(5,2) will have one republican and 2 democrats.

C(5,3) will have all democrats.

C(n,r) = n!/(r!(n-r)!)