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Example text

Then 11. (11. - 1)(11. > (11. - 2)3 and 11. is that natural number for which 504 lies between (11. 3• Note that a problem such as this need not have a solution. For example, the problem of finding an 11. such that C... 3 = 85 has no solution. Solution We list the cubes 33 = 43 = 64, 27, 53 = 125, 68 = 216, 88 = 512, 93 = 729 78 = 343, The only pos81ole solutions of this problem are 11. = 8, 9 and we verify that . CU• 3 = II. Find 11. 9 9·8'7 ~ = 3·4'7 = 84. = 42P... 7• Solution The integer P ...

7. A game wheel is to consist of three relatively fixed rings, each colored in seven differently' colored sections. How many different color arrangements are possible? 17. Distinguishable permutations (FULL COURSE). The study of distinguishable permutations is best begun by a focus of attention on r objects in a sequence of n objects. Let us count the number of permutations derived if we permute these r objects and leave all others fixed. Since the fixed objects might just as well not appear insofar as they affect our count, there are r I such permutations.

N(n - 1) . . (r + 1) groups, of permutations. Each group is such that every permutation in anyone group is obtainable from every other one in the same group by permutating the same set of r prescribed objects,; two permutations in different groups are not obtainable one from the other without permuting some of the remaining n - r objects. Suppose now that the r fixed objects were indistinguishable. Then all the permutations in each of the groups described above cannot be told apart and must be counted as a single permutation.

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