Another method

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The first digit can be chosen in 9 ways, and the next two digits in 8 P2 ways. Then 9 8 P2 504 numbers can be formed. 9 8 7

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1.24. Four different mathematics books, six different physics books, and two different chemistry books are to be arranged on a shelf. How many different arrangements are possible if (a) the books in each particular subject must all stand together, (b) only the mathematics books must stand together

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(a) The mathematics books can be arranged among themselves in 4 P4 4! ways, the physics books in 6 P6 ways, the chemistry books in 2 P2 2! ways, and the three groups in 3 P3 3! ways. Therefore, Number of arrangements 4!6!2!3! 207,360. 6!

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(b) Consider the four mathematics books as one big book. Then we have 9 books which can be arranged in 9! ways. In all of these ways the mathematics books are together. But the mathematics books can be 9 P9 arranged among themselves in 4P4 4! ways. Hence, Number of arrangements 9!4! 8,709,120

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1.25. Five red marbles, two white marbles, and three blue marbles are arranged in a row. If all the marbles of the same color are not distinguishable from each other, how many different arrangements are possible

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Assume that there are N different arrangements. Multiplying N by the numbers of ways of arranging (a) the five red marbles among themselves, (b) the two white marbles among themselves, and (c) the three blue marbles among themselves (i.e., multiplying N by 5!2!3!), we obtain the number of ways of arranging the 10 marbles if they were all distinguishable, i.e., 10!. Then (5!2!3!)N 10! and N 10! > (5!2!3!)

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In general, the number of different arrangements of n objects of which n1 are alike, n2 are alike, . . . , nk are n! c n alike is where n1 n2 n. k n 1!n 2! c n k!

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CHAPTER 1 Basic Probability

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1.26. In how many ways can 7 people be seated at a round table if (a) they can sit anywhere, (b) 2 particular people must not sit next to each other

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(a) Let 1 of them be seated anywhere. Then the remaining 6 people can be seated in 6! the total number of ways of arranging the 7 people in a circle. 720 ways, which is

(b) Consider the 2 particular people as 1 person. Then there are 6 people altogether and they can be arranged in 5! ways. But the 2 people considered as 1 can be arranged in 2! ways. Therefore, the number of ways of arranging 7 people at a round table with 2 particular people sitting together 5!2! 240. Then using (a), the total number of ways in which 7 people can be seated at a round table so that the 2 particular people do not sit together 730 240 480 ways.

Combinations 1.27. In how many ways can 10 objects be split into two groups containing 4 and 6 objects, respectively

This is the same as the number of arrangements of 10 objects of which 4 objects are alike and 6 other objects 10! 10 9 8 7 are alike. By Problem 1.25, this is 210. 4!6! 4! The problem is equivalent to finding the number of selections of 4 out of 10 objects (or 6 out of 10 objects), the order of selection being immaterial. In general, the number of selections of r out of n objects, called the number n of combinations of n things taken r at a time, is denoted by nCr or a b and is given by r c (n r 1) n n(n 1) n Pr n! nCr r!(n r)! r! r! r (a) 7 C4 (b) 6C5 7! 4!3! 7 6 5 4 4! 7 6 5 3 2 1 35. 6! 6 5 4 3 2 6, or 6C5 6C1 6. 5!1! 5! (c) 4 C4 is the number of selections of 4 objects taken 4 at a time, and there is only one such selection. Then 4 C4 Note that formally