Suppose that you have 6 cards, 3 blue and 3 green, and you place them randomly in 6 envelopes, 3 blue and 3 green. What is the probability that exactly two cards will match the color of the envelope?
by 
sjonabro
ANSWERS: 2
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Believe it or not, there are actually only 20 different combinations when you select the cards, since the formula is 6!/3!3!. This is because blue card 1 is identical to blue card 2 and 3, and the same for the green cards. At first thought, the only clear way I can see right now to solve this is to write out all of the combinations, and go through each one, and see how many match exactly two cards. But there must be a more scientific way to do it. Here's the first one: Card Selection BBBGGG has only two matching cards and envelopes with only these other combinations: BGGBBG BGGGBB BGGBGB GBGBBG GBGBGB GBGGBB GGBBBG GGBBGB GGBGBB for a probability of 9/20. And looking at this pattern, my thought is that this is the overall probability, that all the other card selections will match this. "What pattern," you say? Two greens in the first half, and two blues in the second matches the choice pattern BBBGGG nine times. Each selection of cards will have a similar pattern that will meet the criterion.
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Sometimes, it is easier to make all the objects different. So I'll number the cards and envelopes each 1 to 6 in pencil! So we have 6 cards and 6 envelopes, and 6-factorial = 720 ways of arranging the different cards in the envelopes. How many ways of making exactly two cards match? I'll have to put just one blue card in a blue envelope and just one green card in a green envelope. 3 choices for blue card 3 choices for blue envelope 3 choices for green card 3 choices for green evelope Now the non-matching cards have to go in the envelopes of the wrong color. There are two green envelopes to put the two blue cards in, that's two ways round they could go. There are two blue envelopes to put the two green cards, that's two ways round they could go. Total ways of arranging the cards according to the condition is 3*3*3*3*2*2 = 324 Final probability = 324/720 = 54/120 = 9/20
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