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# w11 - there for such a setup 5 How many ways can you split...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Labelled or Not! 1. The 55 students of math 55 have found a bag of 555 gold coins. How many diﬀerent ways can they divide it among themselves if: (a) there are no restrictions on the distribution? (b) each student must get at least one piece of gold? (c) the ﬁercest student gets 300 pieces? 2. (a) How many ways can you arrange n copies of the exact same book (indistinguishable) on a bookcase with r shelves? (b) What if the books are all diﬀerent? 3. How many positive integers less than 1,000,000 have the sum of their digits equal to 9? 4. In bridge, each of 4 players is dealt 13 cards from a standard 52-card deck. How many diﬀerent possibilities are
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Unformatted text preview: there for such a setup? 5. How many ways can you split a set of n elements into 2 disjoint subsets? Note: there is a very short and simple formula for this. Also, YOU need to ﬁgure out what is or isn’t labelled (or distinguishable) in this problem. 6. How many solutions in non-negative integers are there to (a) x 1 + x 2 + x 3 + x 4 + x 5 = 30 and x 1 ≤ 10? (b) x 1 + x 2 + x 3 + x 4 ≤ 30? (Hint: where did x 5 go? I miss it. ..) 7. F Prove that ( n + r +1 r ) = ( n ) + ( n +1 1 ) + ( n +2 2 ) + ··· ( n + r r ) . Hint: the LHS counts the number of ways to split r identical coins among n + 2 people....
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