Quiz 3 Counting: 4.3, 4.4, 4.5

Quiz3: April 27, pm Your answers should be expressed in terms of factorials and/or powers. A lonely gambler sits at a table with one deck of cards (one deck has 52 distinct cards). He draws himself a hand of 5 cards (without replacing the cards). a) How many different hands are possible if the order in which he draws is important? b) How many different hands are possible when the order does not matter? He now adds 9 new identical decks of cards (i.e. the total number of decks is now 10). He again draws himself a hand of 5 cards (without replacing the cards). c) How many different hands are possible with repetition when the order matters? d) How many different hands are possible with repetition when the order does not matter? A second player enters the game. e) What is the total number of ways they can draw 2 hands of 5 cards each, when the order in a hand does not matter (but the players are different)? f) In how many ways can we draw all the cards when the order matters (recall that the decks were indistinguishable)?

Quiz3: April 27, answers a) How many different hands are possible if the order in which he draws is important?  52 different balls in 5 distinguishable slots: P(52,5)=52! / 47! b) How many different hands are possible when the order does not matter?  52 different balls in 5 indistinguishable slots: C(52,5)=52! / 47! 5! c) How many different hands are possible with repetition when the order matters?  52 classes of balls with repetition in distinguishable slots: 52^5. d) How many different hands are possible when the order does not matter?  52 classes of balls with repetition in indistinguishable slots: C(56,5)=56! /51! 5!. e) What is the total number of ways they can draw 2 hands of 5 cards each, when the order does not matter?  Hand player 1: C(56,5). Hand player 2: C(56,5): total (56! / 51! 5!)^2. f) In how many ways can we draw all the cards when the order matters?  Number of permutations of 520 cards when the order matters but 52 sets of 10 cards are indistinguishable: (520)! / (10!)^52