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In a school lottery, you buy a card with printed numbers 1, 2, . . . , 10 and you mark 3 of them. In the end, 3 lucky numbers are selected randomly by the school office, and those who have marked them all will win a prize.

a) What is the chance that you correctly guess exactly two out of three lucky numbers? b) What is the chance that you buy two cards, mark two different triplets and one of your cards wins?

I have the answers to the above questions but I do not understand how we came to them. For (a) we have the solutions as: $C_{3,2}*C_{7,1}/C_{10,3}=7/40$. I do not understand where the $C_{7,1}$ came from. For (b) the answer given is just 1/60 with no formula or explanation. Is (b) 1/60 because if you buy 1 card then the answer is 1/120 and the purchase of the additional card doubles your odds of winning, which leads to the 1/60? This is an old test questions my professor handed out for us to study.

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To guess exactly two lucky numbers, you have two chose two among the three of them (that's $C_3^2$ possibilities), and one among the seven "unlucky" numbers (that's $C_7^1$ possibilities). Then divide that by the total number of possible tickets, that's $C_{10}^3$. – Joel Cohen Sep 23 '12 at 22:36
For (b) you have it just right. As long as the events are mutually exclusive, you can add the probabilities to get the chance of one of them happening. Since you marked different triples, you can't have both cards win. – Ross Millikan Sep 23 '12 at 22:52

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