Probability of selecting at least one from each group probability A team of 5 managers is to be selected from a group of 10 managers - 5 from company A, 3 from company B, and 2 from company C. In how many ways can this be done if the team must contain at least one manager from each company?
I tried selecting one from each company first:
$$
5C1 * 3C1 * 2C1 = 30
$$
Then, I selected 2 from the remaining 7 people:
$$
7C2=21
$$
So I got $ 30 * 21 = 630 $, but the answer is 175. What did I do wrong?
 A: I'm late to the party, but here is why your answer is incorrect:
Lets simplify the question so it's easier to follow. Imagine "Sam" and "Dean" work in company A,"Mike" works in company B and we are asked to select 3 managers in total in a way that it contains at least one manager from each company.
We have three managers in total and we want to select three managers so there is only one way to do so.
Your answer is incorrect because first it picks one manager from company A (2C1 = 2) and one manager from company B (1C1 = 1). Then it picks one manager from the remaining managers (1C1 = 1) and says there are two ways to do this. This method gives you the following selections:
- Picking "Sam" from company A, picking "Mike" from company B and picking "Dean" from company A.
- Picking "Dean" from company A, picking "Mike" from company B and picking "Sam" from company A.

which are the same selection. That's why your answer is always bigger than the correct answer.
A: (5C3 x 3C1 x 2C1) +
(5C2 x 3C2 x 2C1) +
(5C2 x 3C1 x 2C2) + 
(5C1 x 3C2 x 2C2) +
(5C1 x 3C3 x 2C1)
^^^ I think these are different combinations on how this can happen. Add these up.
What you have done is taken the amount of ways you can pick 3 person (1 from each group) where the group sizes are 5,3,2. And then you have multiplied by the amount of ways you can pick 2 people from a group of 7 members. Its nothing close to what the question is asking you.
A: Hint:
Find the number of ways you can select managers from these cases and then add them up.
Case 1 : When  one manager is from Company A and Company B each and rest are from Company C.
Case 2 : When one manager is from A and 2 from B and C each.
like these there will be more cases, find all combination on how you can select managers add them up.
