Example 3. How many words can be formed out of the letters article, so that the vowels occupy the even places?

Here we have to put the 3 vowels in 3 specified places, and the 4 consonants in the 4 remaining places; the first operation can be done in 3 ways, and the second in 4. Hence

In this Example the formula for permutations is immediately applicable, because by the statement of the question there is but one way of choosing the vowels, and one way of choosing the consonants.

EXAMPLES XI. a.

1. In how many ways can a consonant and a vowel be chosen out of the letters of the word courage?

2. There are 8 candidates for a Classical, 7 for a Mathematical, and 4 for a Natural Science Scholarship. In how many ways can the Scholarships be awarded?

3. Find the value of 8P7, 25P, 24C4, 19C14

4. How many different arrangements can be made by taking 5 of the letters of the word equation?

5. If four times the number of permutations of n things 3 together is equal to five times the number of permutations of n-1 things 3 together, find n.

6. How many permutations can be made out of the letters of the word triangle? How many of these will begin with t and end with e?

7. How many different selections can be made by taking four of the digits 3, 4, 7, 5, 8, 1? How many different numbers can be formed with four of these digits?

9. How many changes can be rung with a peal of 5 bells?

10. How many changes can be rung with a peal of 7 bells, the tenor always being last?

11. On how many nights may a watch of 4 men be drafted from a crew of 24, so that no two watches are identical? On how many of these would any one man be taken?

12. How many arrangements can be made out of the letters of the word draught, the vowels never being separated?

13. In a town council there are 25 councillors and 10 aldermen ; how many committees can be formed each consisting of 5 councillors and 3 aldermen ?

14. Out of the letters A, B, C, p, q, r how many arrangements can be made (1) beginning with a capital, (2) beginning and ending with a capital?

15. Find the number of combinations of 50 things 46 at a time. 16. If "C12="Cg, find "C17, 22C

17. In how many ways can the letters of the word vowels be arranged, if the letters oe can only occupy odd places?

18. From 4 officers and 8 privates, in how many ways can 6 be chosen (1) to include exactly one officer, (2) to include at least one officer?

19. In how many ways can a party of 4 or more be selected from 10 persons?

20. If 18C18C, +29 find "C.

21. Out of 25 consonants and 5 vowels how many words can be formed each consisting of 2 consonants and 3 vowels?

22. In a library there are 20 Latin and 6 Greek books; in how many ways can a group of 5 consisting of 3 Latin and 2 Greek books be placed on a shelf?

23. In how many ways can 12 things be divided equally among 4 persons?

24. From 3 capitals, 5 consonants, and 4 vowels, how many words can be made, each containing 3 consonants and 2 vowels, and beginning with a capital?

25. At an election three districts are to be canvassed by 10, 15, and 20 men respectively. If 45 men volunteer, in how many ways can they be allotted to the different districts?

26. In how many ways can 4 Latin and 1 English book be placed on a shelf so that the English book is always in the middle, the selection being made from 7 Latin and 3 English books?

27. A boat is to be manned by eight men, of whom 2 can only row on bow side and 1 can only row on stroke side; in how many ways can the crew be arranged?

28. There are two works each of 3 volumes, and two works each of 2 volumes; in how many ways can the 10 books be placed on a shelf so that volumes of the same work are not separated?

29. In how many ways can 10 examination papers be arranged so that the best and worst papers never come together?

30. An eight-oared boat is to be manned by a crew chosen from 11 men, of whom 3 can steer but cannot row, and the rest can row but cannot steer. In how many ways can the crew be arranged, if two of the men can only row on bow side?

31. Prove that the number of ways in which p positive and n negative signs may be placed in a row so that no two negative signs shall be together is p+iСn.

33. How many different signals can be made by hoisting 6 differently coloured flags one above the other, when any number of them may be hoisted at once?

34. If 28C2: 24C2r-4=225 : 11, find r.

149. Hitherto, in the formulæ we have proved, the things have been regarded as unlike. Before considering cases in which some one or more sets of things may be like, it is necessary to point out exactly in what sense the words like and unlike are used. When we speak of things being dissimilar, different, unlike, we imply that the things are visibly unlike, so as to be easily distinguishable from each other. On the other hand we shall always use the term like things to denote such as are alike to the eye and cannot be distinguished from each other. For instance, in Ex. 2, Art. 148, the consonants and the vowels may be said each to consist of a group of things united by a common characteristic, and thus in a certain sense to be of the same kind; but they cannot be regarded as like things, because there is an individuality existing among the things of each group which makes them easily distinguishable from each other. Hence, in the final stage of the example we considered each group to consist of five dissimilar things and therefore capable of [5 arrangements among themselves. [Art. 141 Cor.]

150. Suppose we have to find all the possible ways of arranging 12 books on a shelf, 5 of them being Latin, 4 English, and the remainder in different languages.

The books in each language may be regarded as belonging to one class, united by a common characteristic; but if they were distinguishable from each other, the number of permutations would be 12, since for the purpose of arrangement among themselves they are essentially different.

If, however, the books in the same language are not distinguishable from each other, we should have to find the number of ways in which 12 things can be arranged among themselves, when 5 of them are exactly alike of one kind, and 4 exactly alike of a second kind: a problem which is not directly included in any of the cases we have previously considered.

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151. To find the number of ways in which n things may arranged among themselves, taking them all at a time, when p of the things are exactly alike of one kind, q of them exactly alike of another kind, r of them exactly alike of a third kind, and the rest all different.

Let there be n letters; suppose p of them to be a, q of them to be b, r of them to be c, and the rest to be unlike.

Let x be the required number of permutations; then if the p letters a were replaced by p unlike letters different from any of the rest, from any one of the x permutations, without altering the position of any of the remaining letters, we could form p new permutations. Hence if this change were made in each of the x permutations we should obtain x × p permutations.

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Similarly, if the q letters b were replaced by q unlike letters, the number of permutations would be

In like manner, by replacing the r letters c by r unlike letters, we should finally obtain x x px | xr permutations.

But the things are now all different, and therefore admit of n permutations among themselves. Hence

which is the required number of permutations.

Any case in which the things are not all different may be treated similarly.

Example 1. How many different permutations can be made out of the letters of the word assassination taken all together?

We have here 13 letters of which 4 are s, 3 are a, 2 are i, and 2 are n. Hence the number of permutations

Example 2. How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1, so that the odd digits always occupy the odd places?

The odd digits 1, 3, 3, 1 can be arranged in their four places in

The even digits 2, 4, 2 can be arranged in their three places in

3

ways 2

.(1).

.(2).

Each of the ways in (1) can be associated with each of the ways in (2).

152. To find the number of permutations of n things r at a time, when each thing may be repeated once, twice,.....up to r times in any arrangement.

Here we have to consider the number of ways in which r places can be filled up when we have n different things at our disposal, each of the n things being used as often as we please in any arrangement.

The first place may be filled up in n ways, and, when it has been filled up in any one way, the second place may also be filled up in n ways, since we are not precluded from using the same thing again. Therefore the number of ways in which the first two places can be filled up is n × n or n2. The third place can also be filled up in n ways, and therefore the first three places in n3 ways.

Proceeding in this manner, and noticing that at any stage the index of n is always the same as the number of places filled up, we shall have the number of ways in which the r places can be filled up equal to n".