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Combinatorics and Factorials

If we have to choose a group of 6 people from a selection of 10, there are

possible ways of making a choice of six. The order of selection doesn't matter here. So that although each of the 10 are distinct, once they are picked they can line up in any order, and the order of lining up does not matter.

Working from first principles we might explain like this. We are going to pick 6 from ten. There are 10 possible choices for the first one, 9 possible choices for the second one, 8 for the third, 7 for the fourth, 6 for the fifth and 5 for the sixth, hence 10*9*8*7*6*5 possible ways for picking six altogether. We can write:


Once we have picked our selection the order in which they are lined up does not matter. For the six we have selected there aredifferent ways of lining them up. To take this into account we can divide (1) by an extra factor of 6! to obtain

possible ways of picking 6 from ten. This is of course the same as calculated above.

In general then there areways of choosingfromwithout regard to order. It should also be noted that