Counting Ill 3.2 Hands with a Full house A Full house is a hand with three cards of one value and two cards of another value. Here {2·,24,2◇,J,J◇} ,7,7} Again, we shift to a problem about sequences. There is a bijection between Full Houses and sequences specifying 1. The value of the triple, which can be chosen in 13 ways. 2. The suits of the triple, which can be selected in(3)ways 3. The value of the pair, which can be chosen in 12 ways 4. The suits of the pair, which can be selected in(2)ways The example hands correspond to sequences as shown below 2,,,◇},J{,◇})…{2·,24,29,J,J} (5,{◇,.9},7,{8,}){5◇,5,59,79,7} By the generalized Product rule, the number of Full houses is Were on a roll-but were about to hit a speedbump 3.3 Hands with Two Pairs How many hands have Two Pairs; that is, two cards of one value, two cards of another value, and one card of a third value? here are examples: {3◇,3·,Q◇,Q,A {99,9◇,59,5晶,K命 Each hand with Two Pairs is described by a sequence consisting of 1. The value of the first pair, which can be chosen in 13 ways 2. The suits of the first pair, which can be selected()ways 3. The value of the second pair, which can be chosen in 12 ways 4. The suits of the second pair, which can be selected in()ways� � � � � � � � � � � � 6 Counting III 3.2 Hands with a Full House A Full House is a hand with three cards of one value and two cards of another value. Here are some examples: { 2♠, 2♣, 2♦, J♣, J♦ 5♦, 5♣, 5♥, 7♥, 7♣ } { } Again, we shift to a problem about sequences. There is a bijection between Full Houses and sequences specifying: 1. The value of the triple, which can be chosen in 13 ways. 2. The suits of the triple, which can be selected in 4 3 ways. 3. The value of the pair, which can be chosen in 12 ways. 4. The suits of the pair, which can be selected in 4 2 ways. The example hands correspond to sequences as shown below: (2, {♠, ♣, ♦} , J, {♣, ♦}) ↔ { 2♠, 2♣, 2♦, J♣, J♦ 5♦, 5♣, 5♥, 7♥, 7♣ } (5, {♦, ♣, ♥} , 7, {♥, ♣}) ↔ { } By the Generalized Product Rule, the number of Full Houses is: 4 4 13 · 12 · 3 · 2 We’re on a roll— but we’re about to hit a speedbump. 3.3 Hands with Two Pairs How many hands have Two Pairs; that is, two cards of one value, two cards of another value, and one card of a third value? Here are examples: { 3♦, 3♠, Q♦, Q♥, A♣ } { 9♥, 9♦, 5♥, 5♣, K♠ } Each hand with Two Pairs is described by a sequence consisting of: 1. The value of the first pair, which can be chosen in 13 ways. 2. The suits of the first pair, which can be selected 4 2 ways. 3. The value of the second pair, which can be chosen in 12 ways. 4. The suits of the second pair, which can be selected in 4 2 ways