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# sol8 - AMS 301.2(Spring 2011 Homework Set 8 Solution Notes...

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AMS 301.2 (Spring 2011) Homework Set # 8 – Solution Notes # 10, 5.3: (a). We first give each person a pear, and now distribute 5 identical apples and 3 identical pears to 3 distinct people: parenleftbigg 5 + 3 - 1 5 parenrightbiggparenleftbigg 3 + 3 - 1 3 parenrightbigg (b). The apples can be distributed 3 5 ways, as each apple has 3 choices. The pears can be split either 1,1,4 (3 ways to choose who gets 4 pears) or 1,2,3 (3! ways to pick who gets 1 or 2 or 3 pears) and 2,2,2. Once we have the split, we can use Theorem 1. The final answer is: 3 5 (3 P (6;1 , 1 , 4) + 3! P (6;1 , 2 , 3) + P (6;2 , 2 , 2)) # 12, 5.3: Here we break into cases depending on k ∈ { 0 , 1 , 2 , 3 } , the number of balls we select out of the pink, lavender, and tan balls. This can be done in ( 3 k ) ways. Then we must select the remaining 8 - k balls out of 3 types (red, white or blue), which can be done in ( 8 - k +3 - 1 8 - k ) ways (there are 8 - k “x’s” and 3 - 1 = 2 “slashes”). k can be any number between 0 and 3, so we sum over those possibilities to get the final answer: 3 summationdisplay k =0 parenleftbigg 3 k

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sol8 - AMS 301.2(Spring 2011 Homework Set 8 Solution Notes...

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