352Assignment1 - x 4 5 Chapter 1 Theoretical Exercise 18 6...

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MATH 352 Assignment #1 Due: Tuesday September 18, 2007 1. Twenty-two individually carved merry-go-round horses are to be arranged in 2 concentric rings on a merry-go-round. In how many ways can this be done if each ring is to contain 11 horses and each horse is to be abreast of another horse? 2. Show that four people of each of n nationalities can stand in a row so that each person stands next to someone of the same nationality, in 12 n (2 n )! ways. 3. Let n = p k 1 1 p k 2 2 ··· p k m m ,where p 1 ,p 2 ,...,p m are distinct prime numbers and k 1 ,k 2 ,...,k m are positive integers. How many ways can n be written as a product of two relatively prime positive integers (ie. as a product of two positive integers which have no common divisor greater than 1)? Assume order does not matter (i.e., 8 · 15 and 15 · 8 are regarded as the same). 4. Let n be a positive integer. Consider the 4-tuples ( x 1 ,x 2 ,x 3 ,x 4 ), where x i ∈{ 1 , 2 ,...,n } for i =1 , 2 , 3 , 4. How many such 4-tuples satisfy x 1 x 2 x 3
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Unformatted text preview: x 4 ? 5. Chapter 1 Theoretical Exercise 18. 6. Two friends go for dinner at a restaurant which has m items on the menu. Each order one item (which may be the same), and if they are still hungry they may then order any selection of items that have not yet been ordered to share. By counting how many diFerent orders they can make, prove the identity: m X k =1 k 2 ± m k ² = ± m + 1 2 ² 2 m-1 Bonus Question: In how many ways can m identical red balls, m identical blue balls, and m identical green balls be distributed amongst boxes labelled 1 , 2 , 3, so that each box gets exactly m balls? * ±or full marks, your answer must be in the form of a simple formula which does not include a series. I’d strongly suggest drawing the solutions for the m = 2 case so that you can check if your formula gives the correct answer. (I did NOT get this right on the ²rst try!)...
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