the remaining letters; and so on; thus there are n3 different permutations of the letters taken two at a time. Similarly by putting successively a, b, c, ...... before each of the permutations of the letters taken two at a time, we obtain n3 permutations of the letters taken three at a time. Thus the whole number of permutations when the letters are taken r at a time will be n".

502. Since the number of combinations of n things taken r at a time must be some integer, the expression

must be an integer. Hence we see that the product of any r successive integers must be divisible by [r. We shall give a more direct proof of this proposition in the chapter on the theory of numbers.

EXAMPLES OF PERMUTATIONS AND COMBINATIONS.

1. How many different permutations may be made of the letters in the word Caraccas taken all together?

2. How many of the letters in the word Heliopolis?

3. How many of the letters in the word Ecclesiastical?

4. How many of the letters in the word Mississippi?

5. If the number of permutations of n things taken 4 together is equal to twelve times the number of permutations of n things taken 2 together; find n.

6. In how many ways can 2 sixes, 3 fives, and 5 twos be thrown with 10 dice?

7. If there are twenty pears at three a penny, how many different selections can be made in buying six-pennyworth? In how many of these will a particular pear occur?

8. From a company of soldiers mustering 96, a picket of 10 is to be selected; determine in how many ways it can be done, (1) so as always to include a particular man, (2) so as always to exclude the same man.

9. How many parties of 12 men each can be formed from a company of 60 men?

10. If the number of combinations of n things r-r together be equal to the number of combinations of n things r + jol together, find n.

11. In how many ways can a party of six take their places at a round table?

12. In how many different ways may n persons form a ring?

13. How many different numbers can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, 9; each of these digits occurring once and only once in each number? How many with the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, on the same supposition?

14. Out of 12 conservatives and 16 reformers how many different committees could be formed each consisting of 4 reformers and 3 conservatives?

15. If there be x things to be given to n persons, shew that n" will represent the whole number of different ways in which they may be given.

16. Suppose the number of combinations of n things taken r together to be equal to the number taken r+1 together, and that each of these equal numbers is to the number of combinations of n things taken r-1 together as 5 is to 4, find the value of n.

17. Given m things of one kind, and n things of a second kind, find the number of permutations that can be formed containing r of the first and s of the second.

18. How many different rectangular parallelepipeds are there satisfying the condition that each edge of each parallele

piped shall be equal to some one of n given lines all of different lengths?

19. The ratio of the number of combinations of 4n things taken 2n together, to that of 2n things taken n together is

1.3.5

(4n-1)

{1.3.5...... (2n-1)}**

20. Out of 17 consonants and 5 vowels, how many words can be formed, each containing two consonants and one vowel?

21. Out of 10 consonants and 4 vowels, how many words can be formed each containing 3 consonants and 2 vowels?

22. Find the number of words which can be formed out of 7 letters taken all together, each word being such that 3 given letters are never separated.

23. With 10 flags representing the 10 numerals how many signals can be made, each representing a number and consisting of not more than 4 flags?

24. How many words of two consonants and one vowel can be formed from 6 consonants and 3 vowels, the vowel being the middle letter of each word ?

25. How many words of 6 letters may be formed with 3 vowels and 3 consonants, the vowels always having the even places?

26. A boat's crew consists of 8 men, 3 of whom can only row on one side and 2 only on the other. Find the number of ways in which the crew can be arranged.

27. A telegraph has m arms, and each arm is capable of n distinct positions; find the total number of signals which can be made with the telegraph, supposing that all the arms are to be used to form a signal.

28. A pack of cards consists of 52 cards marked differently; in how many different ways can the cards be arranged in four sets, each set containing 13 cards?

29. How many triangles can be formed by joining the angular points of a decagon, that is, each triangle having three of the angular points of the decagon for its angular points?

are n points in a plane, no three of which are in ight line with the exception of p, which are all in ight line; find the number of straight lines which ning them.

the number of triangles which can be formed by nts in the preceding question.

are n points in space, of which p are in one plane, o other plane which contains more than three of ny planes are there, each of which contains three

oints in a plane be joined in all possible ways by ht lines, and if no two of the straight lines be arallel, and no three pass through the same point cion of the n original points), then the number of ction, exclusive of the n points, will be

re fifteen boat-clubs; two of the clubs have each e river, five others have two, and the remaining find an expression, for the number of ways in be formed of the order of the 24 boats, observing oat of a club cannot be above the first.

contains 20 books, of which 4 are single volumes, rm sets of 8, 5, and 3 volumes respectively; find ys the books may be arranged on the shelf, the et being in their due order.

e number of the permutations which can be letters composing the word examination taken

en-1 sets containing 2a, 3a, na things

......

be formed by taking a out of the first, 2a out of the second, and so on for each combination, is

Ina {La}"

38. Find the sum of all the numbers which can be formed with all the digits 1, 2, 3, 4, 5, in the scale of 10.

39. The sum of all numbers that are expressed by the same digits is divisible by the sum of the digits.

XXXV. BINOMIAL THEOREM. POSITIVE INTEGRAL EXPONENT.

503. We have already seen that (x + a)3 = x2 + 2xa + a3, and that (x+a)=x+3x2α + 3xα3 +a; the object of the present chapter is to find an expression equal to (x+a)" where n is any positive integer.

504. By ordinary multiplication we obtain

(x+α ̧) (x+α ̧) = x2 + (α ̧ + α ̧) x + α ̧ɑ„ (x + α ̧) (x+α ̧) (x +α ̧) = x2 + (α, ̧ +α ̧+α2) x2

2

(x+α ̧) (x + α ̧) (x + α ̧) (x + α ̧) = x* + (α ̧ +α ̧ + α ̧+α ̧) ×3

1

2

3

Now in these results we see that the following laws hold.

I. The number of terms on the right-hand side is one more than the number of the binomial factors which are multiplied together.

II. The exponent of x in the first term is the same as the number of binomial factors, and in the succeeding terms each exponent is less than that of the preceding term by unity.