what is the combination of six numbers from 1to 72

Question

The question is on combinations

Answer 1

The combination is 72*71*70*69*68*67/(6*5*4*3*2*1)=156,238,908. There are 72 ways of selecting the first number, multiplied by 71 ways of selecting the next and so on. The result has to be divided by the number of ways of arranging 6 different numbers (720), because we are only interested in the combination, and not the order.

Answer 2

Well, you have 72 ways of choosing the first number, and for each of these you have 71 ways of choosing the second (so 72*71*70 and so far), and for each of those pairs you can pick the third number in 70 ways, etc.

So the number of ways of picking an *ordered* sequence of numbers in the desired range is 72 x 71 x 70 x 69 x 68 x 67.

But we only care about unordered sets of six numbers, not a sequence. We are over-counting: every combination of numbers will show up in our process exactly 6!=6x5x4x3x2x1=720 times, because this is just the number of ways to arrange six numbers in some order.

Therefore, the final answer is

72*71*70*69*68*67/(6*5*4*3*2*1)=156,238,908

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