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7 problems match “multiplication principle”. Clear

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1. Distributing distinct problems to friends ●○○○○ 10 ptsAlex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?
2. Pair of aces from two decks ●○○○○ 10 ptsA hand consists of one card drawn from a standard 52-card deck with flowers on the back and one card drawn from a standard 52-card deck with birds on the back. How many such hands consist of a pair of aces?
3. Three-letter words from LOVE ●○○○○ 10 ptsHow many 3-letter arrangements (words need not be meaningful) can we make using the letters of the word LOVE, each letter used at most once?
4. Ace then king without replacement ●○○○○ 10 ptsA standard 52-card deck is shuffled. One card is drawn and not replaced, then a second card is drawn. Find the probability that the first card is an ace and the second card is a king. Give your answer as a decimal rounded to four decimal places.
5. Strings with distinctness constraints ●●○○○ 20 ptsAlec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - the first three letters are pairwise distinct, and so are the last three letters; - the first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string?
6. Gifts to five children with restrictions ●●○○○ 20 ptsFour types of present are available: a candy, a paperback, a dolly, and a Mercedes. Five children, Jackie, Tito, Jermaine, Marlon and Michael, each receive exactly one present. Michael and Jermaine cannot receive the same type of present, Tito and Michael cannot receive the same type of present, and Tito cannot receive a paperback. In how many ways can the presents be given?
7. Each player gets an ace ●●●○○ 50 ptsA standard 52-card deck is shuffled and 2 cards are dealt to each of 4 players. Find the probability that every player receives at least one ace. Give your answer as a decimal rounded to eight decimal places.