Combinations we do not consider the order of succession; so that the combination of two or more things is the same, in whatever order they are taken.

Number of Combinations.

(238.) The number of different combinations that may be formed out of a given number of things, is equal to

the number of permutations that may be formed in
the like case, divided by 1×2×3, and so on,

until the number of factors composing the divisor is equal to the number of things in each combination.

Suppose we wish to find the number of combinations that may be formed out of 4 letters, by taking 3 letters in each combination.

The number of permutations that may be formed of 4 letters, when taken three and three, is

4x3x2=24, (235).

Now there can be but one combination of 3 letters taken all together, while there may be 3×2×1 permutations of those letters, which is 6 permutations for each combination. The required number of combinations is therefore

24(1x2x3)=4.

(239.) The number of different combinations that may be formed by taking one from each of two or more sets of different things, will be found by multiplying together the number of things in the different sets respectively.

For example, to find how many different collections of 3 men might be chosen out of 3 companies containing 4, 5, and 6 men, respectively, by taking one from each.

Each of the 4 men in the first company may be combined, separately, with every one of the 5 men in the second company-which gives

4×5=20 combinations of 2 men in the first two companies.

Again; each of these 20 combinations of 2 men may be combined, separately, with every one of the 6 men in the third company-which gives

4x5×6=120 combinations of 3 men,

with one man from each of the 3 companies in each combination.

EXERCISES..

1. How many different combinations of 2 kinds of metal could be formed of 5 different metals?

Ans. 10.

2. How many different combinations of 4 letters may be formed out of the first 12 letters of the Alphabet?

Ans. 495.

3. How many different collections of 3 persons could be taken out of a company of 13 persons ?

Ans. 296.

4. How many different companies of 4 persons may be taken out of 4 companies containing 5, 7, 8, and 9 persons, respectively? (239). Ans. 2520.

5. How many variations might occur in forming a class of 5 pupils, by taking one from each of 5 other classes consisting of 6, 9, 13, 10, and 12 pupils, respectively?

Ans. 84240.

MISCELLANEOUS EXERCISES

ON THE GENERAL PRINCIPLES OF ARITHMETIC.

1. A person bought of a piece of ground for $73, and sold of his purchase for what it cost him. What part of the whole piece did he sell? and for what sum ?

Ans. ; and $49.

2. An upholsterer bought a quantity of carpeting for $150, at $1 per yard, and sold of it at a profit of of a dollar per yard. What amount of profit did he make on the quantity sold? Ans. $30.

3. A bought of B 34 yards, and of C 46 yards of cloth, at $5 per yard. Having sold of these purchases to D, at a profit of $11 per yard; at what rate must the remainder be sold, that his profit may be $150 on the whole?

Ans. $7 per yard. 4. A merchant sold some remnants of cloth, containing 3 yards, 2 yards, 31 quarters, and 11⁄2 quarters, at the rate of $3 per yard. What did the whole amount to?

Ans. $20.436'. 5. A person who had a journey of 735m. 5fur. to perform, went 13 days at the rate of 40m. 3fur. 20p., and 2 days at the rate of 39m miles per day. What distance then remained to be traveled?

Ans. 130m. 7fur. 20p.

6. A grocer exchanged 29gal. 3qt. 1pt. of brandy, at 433. cents per gallon, for rye at 31 cents per bushel. What quantity of rye did he thus obtain ?

Ans. 41bu. 3pk. 2qt..8pt. 7. A bought of B 42T. 13cwt. 2qr. of iron, of which he

sold 18 T. 14cwt. to C, and the remainder to D. of the whole quantity did he sell to each?

What part

Ans. .438', and .561'.

8. What should be paid for plastering one side of a wall which is 30ft. 10in. long, and 8ft. 9in. high, at 183 cents per square yard? Ans. $5.62'.

9. What would be the cost of excavating a cellar which is 36 feet long, 25ft. 8in. wide, and 8ft. 10in. deep, at the rate of $1.06 per cubic yard? Ans. $321.189'. 10. A reservoir for water is 10ft. 3in. in length, 8 feet in breadth, and 4ft. 11in. in depth. How many barrels of water will it contain ? Ans. 68bar. 22gal. 1qt. 1.7'pt.

11. A farmer wishes to construct a crib which shall con. tain 1000 bushels. What must be the length of the crib, allowing its height to be 7 feet, and its breadth 9ft. 6in. ? Ans. 18.713 feet. 12. A southern merchant purchased in New York, 95yd. 3qr. of calico at 1s. 6d. per yard, 394 yards of cloth at 20s. per yard, and 45yd. 21gr. of silk at 8s. 6d. per yard. What was the amount of his bill in Federal money?

Ans. $165.179'.

13. A company of emigrants has a supply of bread for 25 days, at an allowance of 16. per day. How long would the supply last them at an allowance of day?

14. If $20 will supply a family with barrel, for 3 months; how long will the them with flour at $43 per barrel ?

of a pound per Ans. 41 days. flour, at $5 per

same sum supply

Ans. 39 months.

15. If A could build a wall in 3 days, B in 5 days, and C in 6 days; in what time could the three togther build the wall? Ans. 13 days.

16. If the penny-loaf weighs 12 ounces when wheat is sold at 3s. 4d. per buɛhel; what ought to be the weight of a loaf worth 9d. when wheat is sold at 10s. per bushel ?

Ans. 36 ounces.

17. A, B, and C hire a pasture for $24. A puts in 40 cows for 4 months, B 30 cows for 2 months, and C 36 cows for 5 months; what share of the rent should be paid by each? Ans. $9.60; $3.60; and $10.80.

18. The sum of $2000 is to be divided among three persons in such a manner that the first shall have as much as the second, and the second as much as the third. What are the shares? Ans. $457; $6855; and $8574.

19. In a joint speculation A furnished $5000, B $4000, and C $3500. At the end of 6 months, A withdrew $1500, B $500, and C increased his stock by of its original amount. At the end of 12 months a dissolution occurred, when their profits had amounted to $3765.121; what were the respective shares of profit?

Ans. $1293.07'; $1140.94'; and $1331.10'.

20. How many ounces of gold which is 15 carats fine must be mixed with 3oz. 18 carats fine, and 5oz. 23 carats fine, that the compound may be 20 carats fine? Ans. 14oz.

21. If 18 men build a wall 40 feet long, 3 feet thick, and 16 feet high, in 12 days; how many men will build a wall 360 feet long, 8 feet thick, and 10 feet high, in 60 days?

Ans. 54 men.

22. A vintner has wine at 3s. per gallon, and would mix it with water, so as to obtain 144 gallons which shall be worth 2s. 6d. per gallon. How much wine, and how much water must be taken? Ans. 120gal., and 24 gal.

23. Suppose 97b. of pure gold immersed in a vessel full of water to expel 3lb. of water, 9lb. of pure silver to expel 676. of water, and 97b. of a composition of gold and silver to expel 4lb. of water; what are the respective quantities of gold and silver in the composition?

Ans. 6lb. of pure gold; and 3lb. of pure silver.