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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, Etc.

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.

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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 aaaabbccde, 8 being taken at a time?

Ans. 13.

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

Ans. 2819.

PROBLEM

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.

EXAMPLES.

*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 mul tiplied by the number of quantities in the remaining set, will produce 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 extensive 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?

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Or, 9X9X9X9=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 ?

Ans. 20.

2. A was born when B was 21 years of age: how old will A be when B is 47; and what will be the age of B when A is 69 ?

Ans. A 26, B 8r.

3. What

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, will make the sum 200?

Ans. 77.

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

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 100l. more than part of it, and at the end of 4 years found, that his estate amounted to 103421. 3s. 9d. What had he at first ? Ans. 4000l. 8. Divide 1200 acres of land among A, B and C, so that B may 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.

II. 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 5000l. ?

Ans. 118751.

12. If 1000 men, besieged in a town with provisions for 5 weeks, each man being allowed 16oz. 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?

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Ans.

14. A father dying left his son a fortune,

of which

he ran through in 8 months; of the remainder lasted him a twelve-month longer; after which he had only 410k left. What did his father bequeath him?

Ans. 9561. r3s. 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

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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 to gether. 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 38. rd. a gal. At what rate must I sell a quart of this composition, so as to clear 10 per cent. ? Ans. Is. 3 d. 19. What length must be cut off a board 8 broad, to contain a square foot, or as much as 12 inches in length and 12 in breadth ? Ans. 17in

inches

2 20. What difference is there between the interest of 350l. at 4 per cent. for 8 years, and the discount of the same sum at the same rate and for the same time?

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Ans. 271. 3 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 35. 4d. different. What money did he leave for the widow ? Ans. 6351. 10d. 22. What number is that, from which if you take of, and to the remainder add

be 10?

of

the sum will

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