1×2×3×4×5×6(=the number to be taken at a time)=720 10×9×8x7×6×5(same number from 10)=151200 720)151200(210 the answer.

1440

720

720

2. How many combinations can be made of 2 letters out of 24 letters of the alphabet?

Ans. 276.

3. A general, who had often been successful in war, was asked by his King, what reward he should confer on him for his services; the general only desired a farthing for every file of 10 men in a file, which he could make with a body of 100 men: what is the amount in pounds sterling?

44

Ans. 18031572350l. 9s. 2d.

PROBLEM VI.

To find the number of combinations of any given number of things, by taking any given number at a time; in which there are several things of one sort, several of another, &c.

RULE.

1. Find by trial the number of different forms, which the things, to be taken at a time, will admit of, and the number of combinations in each.

2. Add together all the combinations, thus found, and the sum will be the number required.

EXAMPLES.

1. Let the things proposed be aaabbc; it is required to find the number of combinations, that can be made of every three of these quantities.

2. Let aaabbbcc be proposed; it is required to find the number of combinations of these quantities, taken 4 at a time.

Ans. 10.

3. How many combinations are there in aaaabbcċde, 8 being taken at a time?

Ans. 13.

4. How many combinations are there in aaaaabbbbbccccdd ddeeeefffg, 10 being taken at a time? Ans. 2819.

PROBLEM VII.

To find the compositions of any number, in an equal number of sets, the things themselves being all different.

RULE.*

Multiply the number of things in every set continually together, and the product will be the answer required.

* DEMONSTRATION. Suppose there are only two sets; then it is plain, that every quantity of one set, being combined with every quantity of the other, will make all the compositions of two things, in these two sets; and the number of these compositions is evidently the product of the number of quantities in one set by that in the other.

Again, suppose there are three sets; then the composition of two, in any two of the sets, being combined with every quantity of the third, will make all the compositions of 3 in the 3 sets. That is, the compositions of 2 in any two of the sets, being multiplied by the number of quantities in the remaining set, will pro duce the compositions of 3 in the 3 sets; which is evidently the continual product of all the 3 numbers in the 3 sets. And the same manner of reasoning will hold, let the number of sets be what it will. Q.E.D.

The doctrine of permutations, combinations, &c. is of very ex. tensive use in different parts of the mathematics; particularly in the calculation of annuities and chances. The subject might have been pursued to a much greater length; but what has been done already will be found sufficient for most of the purposes, to which things of this nature are applicable.,

EXAMPLES.

1. Suppose there are 4 companies, in each of which there are 9 men; it is required to find how many ways 4 men may be chosen, one out of each company.

9

9

81

9

729

9

6561

Or, 9×9×9×9=6561 the answer.

2. Suppose there are 4 companies, in one of which there are 6 men, in another 8, and in each of the other two 9; what are the choices, by a composition of 4 men, one out of each company?

Ans. 3888.

3. How many changes are there in throwing 5 dice?

Ans. 7776.

MISCELLANEOUS QUESTIONS.

1.

WHAT difference is there between twice five and

twenty, and twice twenty-five?

of

Ans. 20.

age; how old will

2. A was born when B was 21 years A be when B is 47; and what will be the age of B when A is 60? Ans. A 26, B 81.

3. What number, taken from the square of 48, will leave 16 times 54 ? Ans. 1440.

4. What number, added to the thirty-first part of 3813, Ans. 77. will make the sum 200?

5. The remainder of a division is 325, the quotient 467, and the divisor is 43 more than the sum of both: what is Ans. 390270. the dividend?

*

6. Two persons depart from the same place at the same time; the one travels 30, the other 35 miles a day: how far are they distant at the end of 7 days, if they travel both the same road; and how far, if they travel in contrary directions? Ans. 35, and 455 miles.

7. A tradesman increased his estate annually by 1001. more than part of it, and at the end of 4 years found, What had he at that his estate amounted to 103421. 3s. 9d.

first?

B

may

Ans. 4000l.

8. Divide 1200 acres of land among A, B, and C, so that have 100 more than A, and C 64 more than B. Ans. A 312, B 412, and C 476. 9. Divide 1000 crowns; give A 120 more, and B 95 less, than C.. Ans. A 445, B 230, C 325. 10. What sum of money will amount to 132l. 16s. 3d. in 15 months, at 5 per cent. per annum, simple interest ?

Ans. 1251. 11. A father divided his fortune among his sons, giving A 4 as often as B 3, and C,5 as often as B 6; what was the whole legacy, supposing A's share 50001. ?

Ans. 118751.

12. If 1000 men, besieged in a town with provisions for

5 weeks, each man being allowed 16 oz. a day, were reinforced with 500 men more. On hearing, that they cannot

be relieved till, the end of 8 weeks, how many ounces a day must each man have, that the provision may last that time? Ans. 6 oz.

13. What number is that, to which if of be added, the sum will be 1?

Ans.. of which he

14. A father dying left his son a fortune, ran through in 8 months; of the remainder lasted him twelve months longer; after which he had only 410l. left. What did his father bequeath him? Ans. 9561. 13s. 4d,

15. A guardian paid his ward 3500l. for 2500l. which he had in his hands 8 years. What rate of interest did he allow him? Ans. 5 per cent. 16. A person, being asked the hour of the day, said, the time past noon is equal to of the time till midnight. What was the time? Ans. 20 min. past 5.

17. A person, looking on his watch, was asked, what was the time of the day; he answered, it is between 4 and 5 ; but a more particular answer being required, he said, that the hour and minute hands were then exactly together. What was the time?

Ans. 21 minutes past 4.

18. With 12 gallons of Canary, at 6s. 4d. a gallon, I mixed 18 gallons of white wine, at 4s. 10d. a gal. and 12 gallons of cider, at 3s. 1d. a gal. At what rate must I sell a quart of this composition, so as to clear 10 per cent. ?

Ans. 1s. 34.

inches

19. What length must be cut off a board, broad, to contain a square foot, or as much as 12 inches in length and 12 in breath? *Ans. 1:1 in.

20. What difference is there between the interest of 3501. at 4 per cent. for 8 years, and the discount of the same sum at the same rate and for the same time?

Ans. 271.5s. 21. A father devised of his estate to one of his sons, and of the residue to another, and the surplus to his relict for life; the children's legacies were found to be 2571. 3s. 4d. different. What money did he leave for the widow?. Ans 6351. 10 d.

22. What number is that, from which if you take of and to the remainder add of, the sum will be 10? Ans. 10

191

23. A man dying left his wife in expectation, that a child would be afterward added to the surviving family; and making his will ordered, that, if the child were a son, of his estate should belong to him, and the remainder to his mother; but if it were a daughter, he appointed the mother