Question

From a club of 20 people, in how many ways can a group of three members be selected to attend a conference?​

Answer 1

Answer: 1140

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Reason:

There are 20 ways to pick the first person, 19 for the next, and 18 for the last. We count down by one each time we fill up a slot since we cannot reselect any person more than once.

If order mattered, then we'd have 20*19*18 = 6840 permutations.

However, order does not matter because no member has a special seat or role. The individual members don't matter and instead it's all about the group.

Notice that for any group of 3 people, there are 3*2*1 = 6 ways to arrange such individuals. We have to divide by 6 to go from 6840 permutations to 6840/6 = 1140 combinations.

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Here's a more formulaic approach using the nCr combination formula.

Plug in n = 20 and r = 3

`n C r = (n!)/(r!(n-r)!)\\\\20 C 3 = (20!)/(3!*(20-3)!)\\\\20 C 3 = (20!)/(3!*17!)\\\\20 C 3 = (20*19*18*17!)/(3!*17!)\\\\ 20 C 3 = (20*19*18)/(3!)\\\\ 20 C 3 = (20*19*18)/(3*2*1)\\\\ 20 C 3 = (6840)/(6)\\\\ 20 C 3 = 1140\\\\`