This topic deals with the numbers of ways ways we can pick a selection from a number of possible combinations. For instance, suppose we have 10 people lined up and we have to pick a team of 4.
The number of ways we can pick 4 from 10 is written
or 
Working from first principles we can pick the first from 10, the second from 9, the third from 8, the fourth from 7, hence 10*9*8*7=5040. But the order of the picking will not matter here. The four people can be picked in any order and we have not taken account of this. To take account of this objection we notice that 4 people can be arranged in 4*3*2*1=4!=24 ways, so now we divide 5040 by 24 to get 210.
The order did not matter for the above question, but sometimes the order does matter, For example 10 runners in a race will obviously differentiate between first, second and third place. In this case we find
different possibilities.
Sometimes we have combinations of combinations. Suppose we have 6 men and 5 women. We have to form from these a team of 4 men and 3 women. We can pick the four men in
different ways and the 3 women in
different ways. The choices of men and women are completely independent. INDEPENDENT! That should ring a bell. If probabilities are independent we multiply, and so with combinations. Hence the number of ways in which we can pick four men and three women from 6 men and five women is
Sometimes though, we have to write down list of possible arrangements because not every arrangement is acceptable.
Suppose a committee of 5 people is to be selected from 6 men and 4 women. We are required to find the number of selections which has more men than women.
We could have 5 men and no women:
possible choices.
We could have 4 men and 1 woman:
possible choices.
We could have 3 men and 2 woman:
possible choices.
Hence there are 6+60+120 possibilities.