In Lotto 649, a player selects 6 numbers with replacement from the integers 1 through 49, inclusive. In the actual game, the order of the selection of numbers does not matter. In other words, if the winning lottery number for the week is (2, 17, 29, 8, 10, 3), then any permutations of this set of 6 numbers will also be considered a 'winning ticket'.
a) How many unique lottery numbers are there? Include point-form explanations of how you are counting this.
b) What is the probability of winning the lottery if you know that all 6 numbers of the winning number this week are different?
c) What is the probability of winning the lottery if you know that there are three pairs of doubles in this week's winning number? Include point-form explanations of how you are counting the number of ways to get three pairs of doubles. Note that the number (1, 1, 1, 1, 1, 1) would be considered 6 of a kind, rather than three pairs of doubles. Similarly, (1, 1, 2, 2, 2, 2) would be classified as a double and a quadruple.