Total number of ways to pick 4 cards from 12 unique cards: 12C4
= 12!/(8!*4!) = 495
there are 3 cases:
1. 2 pairs ex. 2,2,1,1
2. 1 pair, 2 individual ex. 4,4,5,3
3. 4 individual ex 1,4,5,6
For case one:
there are 6 pairs, so we will pick 2 of them: 6C2 = 6!/(4!x2!) = 30
We must divide this by 2! since there are two slots for the pairs and without doing this we are double counting solutions, (ex. 4,4,2,2 is the same as 2,2,4,4 so we divide by 2! to account for this).
30/2! = 15
For case two:
we know there are 6 ways to choose a pair, but we still have to pick 2 more cards. Regardless of which card is picked, it will have a matching pair. In order to avoid double counting errors, the fourth card can be picked 8 ways rather than 9.
10*8=80 and divide by 2! since, as in case one, alternate combinations are not considered alternate draws (ex. 2,2,4,1 is the same as 2,2,1,4).
=80/2!=40
so we are left with 6*40= 240 which is the number of card draws that contain exactly 1 pair.
We now have all the figures for each case of draws, so we compute the solution:
case 1 + case 2
240 + 15 = 255
divide the totals for cases 1 and 2 out of the total number of possible draws (12C4 = 495) , and you will have the answer:
255/495 = 17/33 which is answer C
alternatively as mentioned above by another guy whose name I don't have in front of me, 495 minus case 3 (all individual card draws) works too
here's a fast way to do case three:
(12 x 10 x 8 x 6)/4! = 240
I'm skipping numbers because I cannot pick a card's pair since I'm looking for the number of draws that have only individual cards. for example, if the first card drawn is 4, there are in fact 11 cards left, but one of them is another four which does not apply to the case I am trying to solve for, so remove that other four from my drawable cards, and am left with 10. this happens again with the second card, and so on.
I divide by 4! to reflect that for each draw, there are 4! ways to arrange the cards ( ex. 2,3,5,6 ->2,3,6,5 -> 2,5,3,6 -> etc). As illustrated here, regardless of the order, the draw is the same.
495 - 240 = 255 ; 255/495 = 17/33 which is answer C
