Combination Calculator

A combination C(n,r) counts ways to choose r items from n items where ORDER DOES NOT MATTER. Choosing 3 people from 10 for a committee: C(10,3) = 10!/(3! x 7!) = 120 ways. If you chose Alice, Bob, Carol — that is the same as Carol, Bob, Alice in combinations.

A permutation P(n,r) counts ways to arrange r items from n where ORDER MATTERS. Choosing a president, VP, and treasurer from 10 people: P(10,3) = 720. Here, Alice-Bob-Carol and Carol-Bob-Alice are DIFFERENT arrangements. Permutations = Combinations x r!

Combinations: order does not matter (choosing committee members, lottery numbers). Permutations: order matters (ranking, passwords, assigned roles). P(n,r) = n!/(n-r)!. C(n,r) = n!/(r! x (n-r)!). Permutations are always >= combinations (for same n and r).

On a scientific calculator: enter n, press nCr button, enter r, press =. On iPhone calculator (landscape): look for Cr. Most graphing calculators have MATH > PRB menu. In Excel: COMBIN(n,r). In Python: math.comb(n,r) or scipy.special.comb(n,r).

Lottery (pick 6 from 49): C(49,6) = 13,983,816. Probability of winning = 1 in 13,983,816 (about 0.0000071%). Powerball (pick 5 from 69 + 1 from 26): C(69,5) x 26 = 292,201,338. Probability = 1 in 292 million. Combinations make these astronomical numbers precise.