Why do you multiply the two values?

Question

Hi! I am having some trouble doing a combinations problem, which is stated below:

 

How many committees of five people can be selected from 8 men and 8 women if the committee must have 3 men and 2 women?

 

So I understand for the men you would do eight choose three because out of the 8 men, only 3 can be on the committee. Likewise, for the women, you would do eight choose two because out of the 8 total women, only 2 are going to make it on one committee. So once you have those values, which are 28 and 56, my friend in my Algebra IIH class said to multiply those values together. Why do you do that?

 

Then also, why would the answer be 1,568? Is this possible?

 

Thank you,

 

Megan

 

 

Answer 1

Hi Megan - statistics, at least for me, was one of the hardest math subjects for me to make sense of. I hope that this can help...

 

Anytime we are dealing with the number of ways that things can combine, we are dealing with multiplication.  For example, if I want to paint a house with 3 different colors (walls, ceiling, and floor), and have 8 different colors to choose from, the number of PERMUTATIONS (where order matters) would be:

 

               8                   x                  7              x                     6        =336     

(# choices for first color)  (# choices for second color)  (# choices for third color)

 

However, if I wanted to figure out how many COMBINATIONS there are of those 8 different colors (where order doesn't matter, i.e. it doesn't matter what color goes where), I would take that answer I came up with, and divide by the number of ways those 3 colors could be arranged (3x2x1=6)

 

    8 x 7 x 6        =        56 different combinations of those 8 colors.

          6          

 

In the example you gave, we want to come up with a committee of 3 men and 2 women. In this case, choosing persons ABC is the same as choosing persons CAB, so we will be dealing with COMBINATIONS, where we have to divide by the ways those people can be arranged.

 

So, the number of different committees will equal:

 

    #ways men can be combined    x     #ways women can be combined

 

              8 x 7 x 6                       x                 8 x 7

                 3 x 2                                            2 x 1

 

                    56                          x                   28

 

                                               1568

Answer 2

Megan

           Remember you are trying to make up 5 person committees so you can pick one of the 28 and for each one of those you have 56 two person committees so the total number is 28*56=1568.

Another way to visualize it is if you Plot the male groups on the Y axis and the female groups on the X axis each integer grid point (x,y) would be a potential 5 person committee.

         Hope this helps

 

Jim