Problem set 7 (g In how many different ways can 10 indistinguishable balls be placed in four dis tinguishable boxes, such that every box gets 1, 2, 3, or 4 balls? Solution. First, we might as well put 1 ball in every box. Now the problem is to put the remaining 6 balls into 4 boxes so that no box gets more than 3 balls. Now we turn to case analysis. For example, we could put 3 balls in two boxes and o balls in the other two boxes. There are 2l2=6 ways to do this. All cases are listed below distribution of balls of ways 3,3,0,0 6 3.2.1.0 1!=24 2,2,2,0 2,2,1,1 22=6 Thus, there are 6+24+4+4+6= 44 ways in all (h) There are 15 sidewalk squares in a row. Suppose that a ball can be thrown so that it bounces on o, 1, 2, or 3 distinct sidewalk squares. Assume that the ball alw from left to right. How many different throws are possible? As an example, a two-bounce throw is illustrated below Solution 15 15 15 0 Gi) In how many different ways can the numbers shown on a red die, a green die, and a blue die total up to 15? Assume that these are ordinary 6-sided dice Solution. We fall back on explicit enumeration. Let R, G, and b be the values shown on the three dice R=1,B+G=14→0ways R=2,B+G=13→0wav R=3,B+G=12→1way R=4,B+G=11 R=5,B+G=10→3ways R=6,B+G=9 4 way Gj) The working days in the next year can be numbered 1, 2, 3, ., 300. I'd like to avoid as many as possible- - - @ � @ � 4 Problem Set 7 (g) In how many different ways can 10 indistinguishable balls be placed in four dis￾tinguishable boxes, such that every box gets 1, 2, 3, or 4 balls? Solution. First, we might as well put 1 ball in every box. Now the problem is to put the remaining 6 balls into 4 boxes so that no box gets more than 3 balls. Now we turn to case analysis. For example, we could put 3 balls in two boxes and 0 balls in the other two boxes. There are 4! = 6 ways to do this. All cases are listed below: 2! 2! distribution of balls # of ways 4! 3, 3, 0, 0 2! 2! = 6 4! 3, 2, 1, 0 = 24 1! 1! 1! 1! 4! 3, 1, 1, 1 3! 1! = 4 4! 2, 2, 2, 0 3! 1! = 4 4! 2, 2, 1, 1 2! 2! = 6 Thus, there are 6 + 24 + 4 + 4 + 6 = 44 ways in all. (h) There are 15 sidewalk squares in a row. Suppose that a ball can be thrown so that it bounces on 0, 1, 2, or 3 distinct sidewalk squares. Assume that the ball always moves from left to right. How many different throws are possible? As an example, a two­bounce throw is illustrated below.  




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Solution. � � � � � � � � 15 15 15 15 + + + 0 1 2 3 (i) In how many different ways can the numbers shown on a red die, a green die, and a blue die total up to 15? Assume that these are ordinary 6­sided dice. Solution. We fall back on explicit enumeration. Let R, G, and B be the values shown on the three dice. R = 1, B + G = 14 → 0 ways R = 2, B + G = 13 → 0 ways R = 3, B + G = 12 → 1 way R = 4, B + G = 11 → 2 ways R = 5, B + G = 10 → 3 ways R = 6, B + G = 9 → 4 ways (j) The working days in the next year can be numbered 1, 2, 3, . . . , 300. I’d like to avoid as many as possible