With 8 people how many different teams of 3 people can be made? 56
With 5 people how many different teams of 3 people can be made? 10
Without writing out every possible combination, what formula is used to get these answers?
Without going into combinatorial math, you multiply the 3 highest numbers downward, and divide them by the three lowest numbers upward. (876)/(123) = 56.
“In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter.”
And if you want a simple intuitive explanation, imagine first order does matter. There are 8 possible 1st picks, then 7 possible 2nd picks, and 6 possible 3rd picks. So there are 876 possible picks if order matters.
Now how many ways could you arrange to pick a specific pick of each ordering of 3. Any of 3 could be in the first spot. Any of the 2 remaining in the second spot. The last one must go last so 321 orderings for each set of three giving 876/321 possible ways to do it
WITH going into combinatorial math, you can see the above answers are, for n choose m:
n!/((n-m)!m!)
n! - n-factorial - n x (n-1) x (n-2) x 3 x2 x 1
so 2! is 2x1
3! = 3x2x1
etc.
8x7x6 is just 8!/5! (see, the 5x4x3x2x1 cancels out)
If A,B,C are on a team - you can pick A, or B or C first, and so on. So ABC, ACB, BAC, BCA, CAB, CBA are 3! ways to pick the exact same team, since order of picking does not matter, just who’s on the team.
Similarly, how many poker hands are there? 5 cards, order of deal does not matter, out of 52 different cards. 52x51x50x49x48/(5x4x3x2x1) or 52!/(47! x 5!) or 2,598,960