A small employee employs 3 men and 5 women

[saara.sarangi]

A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?

For this question, is there a way to answer it using the slot method? I tried but I can't get the right answer.

[messi10]

hmmm... I think anything that can be solved via the formula can be solved via the slot method. Its possible for one or the other to be more intuitive depending on the question.

Since this is a probability question, we need to find the total number of possible arrangements. Using the the slot method, we need to find ways in which we can arrange the following:
SSSSNNNN

S stands for the selected individuals. N stands for not selected. This is 8! divided by 4! x 4! which is equal to 70.

Now we need to find the number of ways where we have exactly two women i.e. WWMM. To do this, we have to split this part into two. First we need to find ways in which we can select two women out of a total of 5.

Using the slot method, this means
SSNNN

5! divided by 3! x 2! which is equal to 10.

We now need to find the number of ways in which we can select 2 men out of a total of 3:
SSN

3! divided by 2! x 1! which is equal to 3 .

So the total ways in which we can get the following arrangement:
WWMM is 3 x 10 = 30 ways.

Therefore, the probability is 30/70 or 3/7

I am presuming that the reason you got stuck with the slot method is that you were trying to deal with men and women together?

[jlucero]

The difference between the formula and the slot method, is whether we want to count things that don't matter. A simple example is when five people are running a race and we want to award two (similar) medals, we would use the formula: 5!/2!3!, and by the slot method, we would multiply 5 x 4 and then divide by 2! Notice the difference between the two:

Formula
5x4x(3x2x1)
2x1x(3x2x1)

Slot
5x4
2x1

In this problem, here's the formula and the slot method:

Formula
_3! x 5!
2!1! 2!3!

Slot
3x2 x 5x4 = 30
..2 .....2

(periods added for proper spacing)

What we're doing is finding the number of ways to find exactly two men, 3 men x 2 men divided by 2 ways of selecting those two men, and two women, 5 women x 4 women divided by 2 ways of selecting those two women, and then multiplying those together.

Combine this with the total number of ways of selecting 4 people: 9x8x7x6 divided by 4! = 70, and you get your slot method ways of solving this question.