It's hard to answer these broad questions in a few sentences, but I'll give it a shot:

You should use the combinations formula when you are counting GROUPS instead of counting ORDERED LISTS.

Examples:
Ordered lists: Joe plays the role of Sherlock Holmes, Frank plays the role of Dr. Watson.
Groups: Joe and Frank will play the two roles.

In your example, it DOES MATTER who gets what role, so it is an ordered list. If it just asked how many ways you could choose 3 male actors to perform in the play, you would use the combination formula. But in this case, it's not just about choosing actors, it is also about assigning specific roles.

When to add, when to multiply:
When each thing is a complete answer, add them up.
When each thing is a piece of the answer that needs to be combined with other pieces, multiply.

In this case, you need to combine 60 male options with 120 female options to get a complete answer about how the roles can be assigned, so you multiply.

@Darya
To specifically address ur questions.
Lets assume we have 5 males A B C D E and three have to be selected out of them for roles 1 2 3. In this case order is important as having A for role 1, B for role 2, C for role 3 is not the same as having A for 3,B for 2 and C for 1. In such a case you would use permutations. Since, out of 5 we are picking up three and finding different combinations for those 3 we have
5P3=5!/(5-3)!=5!/2!=5*4*3=60

Lets assume the 5 females are F G H I J and the four female roles available are 4 5 6 7. Again in this case order is important as having F play role 4, G play role 5 etc is not the same as having F play 5 and G play 4 etc. Again we need to use the permutation formula. Since out of 5 we are selecting 4. 5P4=5!=5*4*3*2=120

Now, when the question says AND we are supposed to multiply the total number of possibilities. So 60*120=7200. If the question said OR then you would have to ADD the two possibilities. Hope ur clear. Cheers

Darya_NYC wrote:Hello everybody!
I have a question about combination problems. I just do not get some of them.
For example, lets take a problem :

The acting group is putting a play that has 4 female and 3 mail roles. There are 5 women and 5 men. How many ways the roles can be assigned? (it is from The Princeton review)
The explanation says that there is 5*4*3*2*1= 120 ways to assign female roles and 5*4*3= 60 ways to assign male roles. And 60*120=7200 ways to assign all the roles.
I do not understand, when should I use this approach, and when a formula for combinations n!/k!(n-k)! it seems like combination to me and every time I use this formula, it is not a right way to me. And also how do I know, when I suppose to plus the results (like 60 and 120) or multiply them?
Thank you and I hope someone can explain this to me) let me know, if it sounds confusing)

I think of this as a problem with "multiple buckets".
We have a bucket of men and a bucket of women.
We have to choose from each bucket.
Given a problem with multiple buckets:

1. Count the number of options from each bucket.
2. Multiply the results.

Note that to ASSIGN means to ARRANGE.

Bucket of women:
5 women, 4 roles.
The number of ways to ARRANGE 4 elements from 5 choices = 5*4*3*2 = 120.

Bucket of men:
5 men, 3 roles.
The number of ways to ARRANGE 3 elements from 5 choices = 5*4*3 = 60.

To combine the choices from each bucket, we multiply:
120*60 = 7200.

Other problems with "multiple buckets":

https://www.beatthegmat.com/combination- ... 85034.html

https://www.beatthegmat.com/combinatorics-t71136.html

Thanks guys! It looks a little more clear to me.
We ll see how i ll do when I will get the problem