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# In the lottery there are 49 balls. How many different combinations are there of a drawing of 6 balls?

Question #51851. Asked by hvj. (Oct 21 04 12:58 AM)

peasypod

Choosing 6 objects from 49 possibilities gives....
49!/(49-6)!6! = 13,983,816 combinations

where the "factorial" is defined by
x! = x*(x-1)*(x-2)*.....*3*2*1

for a calculator of combinations see http://www.wcrl.ars.usda.gov/cec/java/comb.htm

 Oct 21 04, 2:14 AM
gmackematix

Hvj may have given two examples of consecutive balls but only asked how many "different combinaions" there were. Peasy has the correct answer, now let's try and put it as clearly as possible.

For each of the 49 ways to pick the first ball, there are 48 ways to pick the second, leaving 47 ways to pick the third ball and so on. The total number of ways of picking six balls is out is therefore equal to 49x48x47x46x45x44.

However picking 1,2,3,4,5,6 is no different in lottery terms to picking out 6,3,1,4,2,5 or 5,1,3,2,6,4. So above we have counted every group of six balls a number of times. This number is the number of different ways of arranging six balls in order. If we take 10,15,20,25,30,35 as an example. In any arrangement 10 must come first, second, third, fourth, fifth or sixth. This leaves five places for 15, four for 20 and so on. So the number of ways six balls can be arranged is 6x5x4x3x2x1 = 720.
As each group of six balls in the answer above we can divide by 720.

A calculator shows that 49x48x47x46x45x44/720 = 13,983,816.

 Oct 21 04, 5:21 PM

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