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Quiz of the Month (April 2001)

Hector C. Parr

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SOLUTION TO LAST MONTH'S QUIZ

         1) 20 mph

         2) 37, 51, 60, 65, 87

         3) (a) 9    (b) 265

Notes 1). Suppose normal speed = v mph Then normal time = 5/v hrs. So 5/(v + 10) = 5/v - 1/12 Solving, v = 20 (or v = -30 ! ) 2). By intelligent trial and error. 3). The following is a general solution: Let f(n,m) represent the number of arrangements of the numbers 1 to n with exactly m in their correct positions. It is easy to show that f(n,m) = nCm f(n-m,0) (where nCm is the number of selections of n objects takem m at a time). This enables the following table to be built one column at a time, using also the fact that the total for each column must be factorial n. n = 1 2 3 4 5 6 (etc) m = 0 0 1 2 9 44 265 1 1 0 3 8 45 264 2 1 0 6 20 135 3 1 0 10 40 4 1 0 15 5 1 0 6 1 Alternatively, solution by computer is possible.

THIS MONTH'S QUIZ

  1. Two oranges and three lemons cost 119 pence. If an
     orange costs 7p more than a lemon, what does a
     lemon cost? 

  2. Andrew and Barry together have 29 in their pockets,
     Barry and Charles together have 40, while Charles
     and David together have 65. David has twice as much
     as Andrew. How much has Barry? 

  3. A wine-glass contains 10 cl of wine, and a water-jug
     contains 100 cl of water. If 1 cl of water is taken
     from the jug and well mixed with the wine, and then
     1 cl of the mixture is returned to the water-jug, which
     is greater, the volume of water in the wine or wine in
     the water?

***

(c) Hector C. Parr (2001)


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