Permutations and Combinations

This topic deals with the numbers of ways ways we select from a number of group of objects. Suppose we are to pick 4 people from a group of 10.

The number of ways we can choose 4 from 10 (We often say'10 choose 4') is writtenor

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. The order of the picking will not matter here. The four people can be picked in any order and 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 problem, but sometimes it does matter. For example if there are 10 runners in a race will obviously matter whose comes first, second and third. In this case we finddifferent possibilities.

Sometimes we have combinations of combinations. Suppose we have 6 men and 5 women. We have to pick from these a team of 4 men and 3 women. We can pick the men indifferent ways and the women in different ways. The choices of men and women are completely independent. If probabilities or combinations are independents are independent we multiply. Hence the number of ways of picking four men and three women from six men and five women is

Sometimes, we have to write down list of possible arrangements because not every arrangement is acceptable.

Suppose a committee of five people is to be selected from six men and four women. We are required to find the number of possibilities with more men than women.

We could have five men and no women:possible choices.

We could have four men and one woman:possible choices.

We could have three men and two woman:possible choices.

Hence there are 6+60+120 possibilities.