Possibility Question with permutation or combinations?

Puzzling: The total ways to pick *any* 6 cards would be

The total ways to pick *any* 6 cards would be: C(52,6). The ways to pick a hand with *no* diamonds would be C(39,6). If you subtract, you get the total ways to pick a hand with at least one diamond. C(52,6) - C(39,6) = 17,095,897 ways

Captain Matticus, LandPiratesInc

At least 1 card is a diamond. There's another way to look at this: How many ways can we select 6 cards and none are diamonds? 39/52 for the first card 38/51 for the second card 37/50 36/49 35/48 34/47 or (39 * 38 * 37 * 36 * 35 * 34) / (52 * 51 * 50 * 49 * 48 * 47) => (39! / 33!) / (52! / 46!) => 39! * 46! / (33! * 52!) That's the probability of no diamonds. Subtract that from 1 (since 1 represents all probabilities) 1 - (39! * 46! / (33! * 52!)) => (33! * 52! - 39! * 46!) / (33! * 52!) => 0.8397416413373860182370820668693... 83.97% chance of not pulling a diamond EDIT: Sorry, misread that. So, just multiply that value by the number of 6 card hands 52C6 => 52! / (6! * (52 - 6)!) => 52! / (6! * 46!) (52! / (6! * 46!)) * (33! * 52! - 39! * 46!) / (33! * 52!) => (33! * 52! - 39! * 46!) / (6! * 33! * 46!) => 33! * 46! * (52 * 51 * 50 * 49 * 48 * 47 - 39 * 38 * 37 * 36 * 35 * 34) / (6! * 33! * 46!) => (52 * 51 * 50 * 49 * 48 * 47 - 39 * 38 * 37 * 36 * 35 * 34) / (6 * 5 * 4 * 3 * 2) => 6 * (52 * 51 * 50 * 49 * 8 * 47 - 39 * 38 * 37 * 6 * 35 * 34) / (6 * 5 * 4 * 3 * 2) => (52 * 51 * 50 * 49 * 8 * 47 - 39 * 38 * 37 * 6 * 35 * 34) / (5 * 4 * 3 * 2) => 5 * (52 * 51 * 10 * 49 * 8 * 47 - 39 * 38 * 37 * 6 * 7 * 34) / (5 * 4 * 3 * 2) => (52 * 51 * 49 * 47 * 10 * 8 - 39 * 38 * 37 * 34 * 7 * 6) / (4 * 3 * 2) => 13 * (4 * 51 * 49 * 47 * 10 * 8 - 3 * 38 * 37 * 34 * 7 * 6) / (4 * 3 * 2) => 13 * 4 * (51 * 49 * 47 * 10 * 8 - 19 * 37 * 17 * 7 * 6 * 3) / (4 * 3 * 2) => 13 * (51 * 49 * 47 * 10 * 8 - 37 * 19 * 17 * 7 * 6 * 3) / (3 * 2) => 13 * 3 * (17 * 49 * 47 * 10 * 8 - 37 * 19 * 17 * 7 * 6) / (3 * 2) => 13 * 17 * (49 * 47 * 10 * 8 - 37 * 19 * 7 * 6) / 2 => 13 * 17 * (49 * 47 * 10 * 4 - 37 * 19 * 7 * 3) => 221 * (49 * 47 * 10 * 4 - 37 * 19 * 7 * 3) 49 * 47 => (48 + 1) * (48 - 1) => (48^2 - 1^2) => ((50 - 2)^2 - 1) => (2500 - 200 + 4 - 1) => 2303 37 * 19 => 37 * 20 - 37 => 740 - 37 => 703 221 * (49 * 47 * 10 * 4 - 37 * 19 * 7 * 3) 221 * (2303 * 4 * 10 - 703 * 21) => 221 * (9212 * 10 - (14060 + 703)) => 221 * (92120 - 14763) => 221 * (78120 - 763) => 221 * (78120 - (120 + 643)) => 221 * (78120 - 120 - 643) => 221 * (78000 - 643) => 221 * 77357 77357 * 2 = 154714 77357 * 1 = 77357 77357 * 221 => 77357 * (200 + 20 + 1) => 15471400 + 1547140 + 77357 10^7 * (1) + 10^6 * (5 + 1) + 10^5 * (4 + 5) + 10^4 * (7 + 4 + 7) + 10^3 * (1 + 7 + 7) + 400 + 140 + 357 10,000,000 + 6,000,000 + 900,000 + 180,000 + 15,000 + 540 + 357 16,900,000 + 180,000 + 15,000 + 897 => 17,080,000 + 15,897 => 17,095,897 Calculators are awfully swell, ain't they?