Problem Set 8 hus, by the Pigeonhole Principle, f is not injective; that is, there exist distinct el- ements a1, a2 E A such taht f(ai=f(a2). In other words, there are two different sequences of button presses that produce the same configuration Problem 2. Suppose you have five 6-sided dice, which are colored red, blue, green, white, and black. a roll is a sequence specifying a value for each die. For example, one roll is red green blue white black For the problems below, you do not need to simpify your answers, but briefly explain your reasoning (a) For how many rolls is the value on every die different? Example: (1, 2, 3, 4, 5)is a roll of this type, but(1,1, 2, 3, 4)is not Solution. The number of such rolls is6·5·4·3.2 (b) For how many rolls do two dice have the same value and the remaining three dice all have different values? Example:(6, 1,6, 2, 3)is a roll of this type, but(, 1, 2, 2, 3)and(4, 4, 4, 5, 6)are not Solution. There are possible pairs of rolls that might have the same value and 6 possibilities for what this value is. There 5. 4.3 possible distinct values for the remaining three rolls. So the number of rolls of this type is 6·5·4·3 (c) For how many rolls do two dice have one value two different dice have a second value, and the remaining die a third value? Example:(6, 1, 2, 1, 2)is a roll of this type, but (4, 4, 4, 4, 5)and(5, 5, 5, 6, 6)are not Solution. There are (2)sets of two values that might be duplicated. There are( 2) rolls where larger duplicated value may come up and ) remaining rolls where the smaller duplicated value may come up. There is only 1 remaining roll where the nonduplicated value may then come up, and 4 remaining values it could take. So, the number of rolls of this ty (( m problem concerns seven card hands dealt from a regular 52-card deck.� � � � � � � � � � � � � � � � 2 Problem Set 8 Thus, by the Pigeonhole Principle, f is not injective; that is, there exist distinct el￾ements a1, a2 ∈ A such taht f(a1) = f(a2). In other words, there are two different sequences of button presses that produce the same configuration. Problem 2. Suppose you have five 6­sided dice, which are colored red, blue, green, white, and black. A roll is a sequence specifying a value for each die. For example, one roll is: ( 3 , 1 , 4 , 1 , 5 ���� ���� ���� ���� ����) red green blue white black For the problems below, you do not need to simpify your answers, but briefly explain your reasoning. (a) For how many rolls is the value on every die different? Example: (1, 2, 3, 4, 5) is a roll of this type, but (1, 1, 2, 3, 4) is not. Solution. The number of such rolls is 6 5 4 3 2 · · · · (b) For how many rolls do two dice have the same value and the remaining three dice all have different values? Example: (6, 1, 6, 2, 3) is a roll of this type, but (1, 1, 2, 2, 3) and (4, 4, 4, 5, 6) are not. Solution. There are 5 2 possible pairs of rolls that might have the same value and 6 possibilities for what this value is. There 5 4 3 · · possible distinct values for the remaining three rolls. So the number of rolls of this type is 5 6 5 4 3 2 · · · · (c) For how many rolls do two dice have one value, two different dice have a second value, and the remaining die a third value? Example: (6, 1, 2, 1, 2) is a roll of this type, but (4, 4, 4, 4, 5) and (5, 5, 5, 6, 6) are not. Solution. There are 6 2 sets of two values that might be duplicated. There are 5 2 rolls where larger duplicated value may come up and 3 2 remaining rolls where the smaller duplicated value may come up. There is only 1 remaining roll where the nonduplicated value may then come up, and 4 remaining values it could take. So, the number of rolls of this type is: 6 5 3 4 2 · 2 · 2 · Problem 3. This problem concerns seven card hands dealt from a regular 52­card deck