Combinations or permutations A company has $12$ construction workers. The manager plans to assign $3$ workers to site A, $4$ to job site B, and $5$ to job site C. In how many different ways can the manager make this assignment? 
I tried to approach this question by multiplying $\binom{12}{3}$ times $\binom{9}{4}$ times $\binom{5}{5}$. 
I don't understand how to approach this question, I hope I can get a clear explanation on how to use combination (permutation as well!) Thanks in advance.
 A: I don't understand how to approach this question
-> Yes you do !
I tried to approach this question by multiplying 12C3 times 9c4 times 5C5
-> Excellent, you have the right approach and the right formula.
A: You have to choose between $12$ construction workers. So let's say $N=12$.
Then, how many possibilities there are to choose $3$ workers for site $A$?
The answer is $$\binom{12}{3}$$Now you have to choose another group of $4$ people to work on site $B$. You have $$\binom{9}{4}$$ possibilities. That's why you've already chosen $3$ people working on site $A$. Now, the remaining $5$ automatically go on site $C$.
So the answer is $$\binom{12}{3} \binom{9}{4}$$
Little explanation:
In how many ways can you choose a subgroup of $k$ elements between $N$? The answer is $\binom{N}{k}$.
A: Your approach is correct. You first choose a group of $3$ workers out of $12$ in $\binom{12}{3}$ ways. Then you choose a group of $4$ workers out of $12-3=9$ workers in $\binom{9}{3}$ ways. The remaining $12-3-4=5$ are assigned to the job $C$. All together we get
$$\binom{12}{3}\cdot \binom{9}{4}=\frac{12!}{3!4!5!}.$$
Note that that final formula depends only on the total number of workers and the cardinalities of the groups. It does not depend on the order of choice. For example you can also first choose the workers for site $C$, then for site $B$ and finally site for $A$. The result is the same:
$$\binom{12}{5}\cdot \binom{7}{4}=\frac{12!}{3!4!5!}.$$
A: Combination suffice when we don't need to arrange things. In your case once you choose your workers for diffrent jobs then there is no need to.arrange them as if they will do the same job no matter in what order.
As far as choosing of workers  is concerned, you have got it right. 
