I need help with a question, I have no idea where to start. If someone can help me solve it, and explain it at the same time that would be great. Here it is:
In a lucky draw, there are 20 names in a box, and 3 are to be taken out. Find the number of ways in which those three names can be taken out.
For the first name, you have $20$ options, for the second name, you only have $19$ options, and for the third name, your are left with $18$ options. So there are $$ 20*19*18 = 6840$$ ways to draw $3$ names out of a box with $20$ names.
On the other hand, if you are interested in finding out all possible combinations of names if you draw all three names at the same time, then the possible combinations of names is equivalent to the possible subsets of $\{1,...,20 \}$ of power $3$, which is given by
$$ {{20} \choose {3}} = \frac{20!}{3! 17!} = 1140. $$