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PERMUTATION. Permutation is the method of finding how many changes may be made, in the order in which things succeed each other.

What number of permutations may be made on the letters A and B? They may be written A B, or B. A.

What number on the letters A B C ?
Placing A first, A B C, or A C B.
Placing B first, B A C, or BC A.
Placing C first, C A B, or C B A.

From these examples it will be seen, that of two things, there may be 2 changes, (1 X2=2,) and of 3 things there may be 6 changes, (ix2X3=6.)

Hence, to find the number of different changes, or permutations, of which any number of different things are capable,

Find the continual product of the natural series of num. bers, from 1 to the given number.

1. Four gentlemen agreed to remain together, as long as they could arrange themselves differently at dinner. How many days did they remain ?

A. 24 days, 2. 10 gentlemen made the same agreement, but they all died before it could be fulfilled. The last survivor lived 53 yrs. 98 days, after the agreement. How much did the bargain then want of being

fulfilled, allowing 365 days to the year?

A. 9,888 yrs. 237 d. 3. How many years will it take to ring all the possible changes on 12 bells, supposing that 10 can be rung in a minute, and that the year contains 365 d. 5 h. 49 m?

A. 91 yrs. 26 d. 22 h. 41 m. 4. How many variations may there be in the position of the nine digits ?

Ans. 362880. 5. A man bought 25 cows, agreeing to pay for them 1 cent for every different order in which they could all be placed; how much did the cows cost him?

Ans. $155112100433309859840000.

What is permutation? What is the rule ?

MISCELLANEOUS EXAMPLES. Many of these sums are designed for mental exercise. In solving the first 50, the pupil should not be allowed to use the slate.

1. If two men start from the same place and travel in opposite directions, one at the rate of 48 miles an hour, and the other at the rate of 3 miles an hour, how far will they be apart in 6 hours ?

2. If 6 bushels of oats will keep 3 horses a week, how many bushels will be required to keep 12 horses the same time ?

3. If you give 5 men 34 bushels of corn apiece, how much do you give the whole ?

4. If 8 dollars worth of provisions will serve 9 men 5 days, how many days will it serve 12 men ? how many days would it serve 3 men ?

5. If $6 worth of provision will serve 5 men 8 days, how many days would it serve 9 men ? how many days would it serve 3 men ?

6. If $12 worth of provision would serve 5 men 7 days, how many men would it serve 9 days ?

7. If one peck of wheat afford 9 six penny loaves, how many ten penny loaves would it afford ?

8. If a man paid $60 to his laborers, giving to every man 9d. and to every boy 3d. if the men and boys were equal in number, how many were there of each?

9. Two men bought a barrel of flour together, one paid $3 and the other paid $5; what part of the whole did each pay, and what part of the barrel ought each to have ? 10. Three men bired a field together, A paid $7,

B paid $3, and C paid $3, what part of the whole did each pay, and what part of the produce ought each to have ?

11. Three men bought a lottery ticket together, A paid $6, B paid $4, and C paid $10. They drew a prize of $150, what was each man's share ?

12. Three men hired a pasture together for $60. A put in 2 horses, B 4 horses, and C6 horses, how much ought each to pay ?

13. Three men commenced trade together, and advanced money in this proportion-For every $5 that A

money in

put in, B put in 3, and C put in $2, they gained $100, what was each man's share ?

14. Two men hired a pasture for $32. A put in 3 sheep for 4 months, and B put in 4 sheep for 5 months, how much ought each to pay ?

Note. 3 sheep for 4 months is the same as 12 sheep for one month, and 4 sheep for 5 months is the same as 20 sheep for one month.

15. A and B traded together and invested money in the following proportions, A put in 10 for 2 months, and B put in $5 for 3 months. They gained $70; what was each man's share ? 16. Three men traded in

company, and

put

in the following proportions. A put in 4 dollars as often as B put in 3, and as often as C put in 2. A's money was in 2 months, B's 3 months, and C's 4 months. They gained $100 ; what was each man's share ?

17. Two men traded in company. A put in $2 as often as B put in $3. A's money was employed 7 months, and B’s 5 months. They gained 58 dollars. What was each man's share ?

18. If A can do } of a piece of work in 1 day, and B can do į of it in one day, how much would both do in a day? How long would it take them both together to do the whole ?

19. If 1 man can do a piece of work in 2 days, and another in 3 days, how much of it would each do in a day ? How much would both together do ? How long would it take them both to do the whole ?

20. A cistern has 2 cocks; the first will fill it in 3 hours, the second in 6 hours; how much of it would each fill in an hour? How much would both together fill ? How long would it take them both to fill it ?

21. A man and his wife found by experience, that, when when they were both together, a bushel of meal would last them only 2 weeks; but when the man was gone,

it would last his wife 5 weeks. How much of it did both together consume in 1 week? What part did the woman alone consume in 1 week? What part did the man alone consume in 1 week? How long would it last the man alone ?

22. If 1 man could build a piece of wall in 5 days, and another man could do it in 7 days, how niuch of it would each do in 1 day? How many days would it take them both to do it?

23. A cistern has 3 cocks; the first would fill it in 3 hours, the second in 6 hours; the third in 4 hours ; what part of the whole would each fill in 1 hour ? and how long would it take them all to fill it, if they were all running at once ?

24. A and B together can build a boat in 8 days, and with the assistance of C they can do it in 5 days; how much of it can A and B build in 1 day? How much of it can A, B, and C, build in 1 day? How much of it can C build alone in 1 day? How long will it take C to build it alone ?

25. Suppose I would line 8 yards of broadcloth that is 1} yards wide, with shalloon that is of a yard wide; how many yards of the shalloon will line 1 yard of the broad. cloth ? How many yards will line the whole ?

26. If 7 yards of cloth cost 13 dollars, what will 10 yards cost ?

27. If the wages of 25 weeks come to 75 dollars, what will be the wages of seven weeks ?

28. If 8 tons of hay will keep 7 horses three months, how much will keep 12 horses the sarne time ?

29. If a staff 4 feet long cast a shadow 6 feet long, what is the length of a pole that casts a shadow 58 feet at the same time of day?

30. If a stick 8 feet long cast a shadow 2 feet in length, what is the height of a tree which casts a shadow 42 feet at the same time of day ?

31. A ship has sailed 24 miles in 4 hours; how long will it take her to sail 150 at the same rate ?

32. 30 men can perform a piece of work in 20 days ; how many men will it take to perform the same work in 8 days!

33. 17 men can perform a piece of work in 25 days ; in how many days would 5 men perform the same work ?

34. A hare has 76 rods the start of a greyhound, but the greyhound runs 15 rods to 10 of the hare; how many rods must the greyhound run to overtake the hare ?

35. A garrison has provision for 8 months, at the rate of 15 ounces per day; how much must be allowed per day, in order that the provision may last 11 months ?

36. If 8 men can build a wall 15 rods in length in 10 days, how many men will it take to build a wall 45 rods in length in 5 days?

37. A man being asked the price of his horse, answered, that his horse and saddle together were worth 100 dollars; but the horse was worth 9 times as much as the saddle. What was each worth ?

38. A man having a horse, a cow, and a sheep, was asked what was the value of each. He answered that the cow was worth twice as much as the sheep, and the horse 3 times as much as the sheep, and that all together were worth 60 dollars. What was the value of each ?

39. If 80 dollars worth of provision will serve 20 men 24 days, how many days will 100 dollars worth of provi. sion serve 30 men ?

40. The third part of an army was killed, the fourth part taken prisoners, and 1000 fled; how many were in this army?

This, and the following 10 questions, are usually classed under the rule of Position, but they may be solved in a much more simple and easy manner. Thus, += 1, of the army. Now as there are 12 twelfths in the whole, 1000 must be the remaining 5 twelfths. If 1000 is 5 twelfths of the ar. my, 1 fifth of 1000, or 200, will be 1 twelfth ; and if 200 is I twelfth, the whole, or 12 twelfths will be 12 times as much, or 2400.

41. A farmer being asked how many sheep he had, an. swered, that he had them in 4 pastures; in the first he had

} of his flock; in the second į; in the third ;; and in the fourth 15; how many sheep had he?

42. A man driving his geese to market, was met by another, who said, good morrow, master, with your hun. dred geese ; says he, I have not a hundred; but if I had half as many more as I now have, and two geese and a half, I should have a hundred; how many had he ?

43. What number is that, to which if its half be added, the sum will be 60 ?

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