Therefore, I must take 10 gallons each of the two sorts, which are worth 1 dollar 26 cents, and 2 dollars 12 cents a gallon.

20. How much gold of 14 and 16 carats fine must be mixed with 6 ounces of 19, and 12 oz. of 22 carats fine, that the composition may be 20 carats fine?

21. A silversmith has silver of 6, 7, and 9 ounces fine, which he wishes to mix with 9 ounces of 10 ounces fine, and 9 ounces of pure silver, to make a mass, that shall be 8 ounces fine. How much of each of the three first must he take?

22. A lady purchases 7 yards of calico at 22 cents a yard, and 7 yards at 20 cents a yard, and wishes to know how many yards of two other kinds, one at 16 cents and the other at 17 cents a yard, she must purchase, to make the average price of the whole 18 cents a yard. Find the two quantities.

CASE IV. When the whole compound is limited to a certain quantity.

RULE. Find an answer, as in Case 1, by alligating; then, as the sum of the quantities thus found, is to the given quantity, so is the quantity of each ingredient found by alligating, to the required quantity of it.

23. A goldsmith has gold of 15, 17, 20, and 22 carats fine; and would melt together of all these sorts so much, as to make a mass of 40 ounces 18 carats fine. How much of each sort is required?

Ans. 16 oz. of 15; 8 oz. of 17; 4 oz. of 20; and 12oz.

of 22 carats fine.

24. Having three sorts of raisins at 9, 12, and 18 cents

a pound, what quantity of each sort must I take, to fill a cask containing 210 pounds, that its contents may be worth 14 cents a pound?

25. Of four different kinds of apples at 31, 37, 46, and 74 cents a bushel, what quantity of each must be taken, to fill a bin containing 9 bushels, to make its contents worth 50 cents a bushel?

XXXVI.

PERMUTATIONS.

PERMUTATION—which is also called variation—means the different ways in which the order or relative position of any given number of things may be changed. The only object to be regarded in Permutation, is the order in which the things are placed; for no two arrangements are to have all the quantities in the same relative position.

For example, two things, a, and b, are capable of only two changes in their relative position, viz. ab, ba; and this number of changes is expressed by 1X2; but three things, a, b, and c, are capable of six variations, viz. a b c, a cb, ba c, bca, ca b, c b a, and this number of permutations is expressed by 1×2×3; and four things, a, b, c, and d, are capable of 24 variations, viz. a b c d, abd c, a c bd, a cd b, ad bc, adc b; ba cd, ba C, b cad, b c d a, b d a c, bdca; cab d, cad b, cbad, cbd a, cda b, c d ba; da u c, d a c b, db a c, d be a, dc ab, dc ba; and this number of permutations is expressed by 1X2×3×4.

d

In like manner, when there are 5 things, every four of them, leaving out the 5th, will have 24 variations; consequently by taking in the 5th, there will be 5 times 24 variations.

PROBLEM I. To find the number of permutations that can be made of any given number of things, all different from each other.

RULE. Multiply the terms of the natural series of

numbers, from 1 up to the given number of things, continually together, and the product will be the answer.

1. How many changes can be made in the order of the six letters, a b c d e f?

2. How many changes may be rung on seven bells? 3. Five gentlemen agreed to board together, as long as they could seat themselves every day in a different position at the dinner table. How long did they board together?

4. How many changes may be made in the order of the words in the following verse? Proci tot tibi sunt, virgo, quot sidera coelo.

5. How many different sums of dollars can be expressed by the nine digits, without using any one of them more than once in the same sum ?

6. How many different arrangements may be made in seating a class of 20 scholars ?

7. A gentleman, who had a wife and eight daughters, one day said to his wife, that he intended to arrange the family in a different order every day at the dinner table, and that he would never give one of his daughters in marriage, till he had completed all the different arrangements of which the family was capable. How many years from that day must elapse, before either of his daughters can be married?

When several of the things are of one sort, and several of another, &c. the changes that can be made upon the whole is not so great, as when all the things are different. For instance, we have seen that the letters a b с admit of six variations; but, if two of the quantities be alike, as a a b, the six variations are reduced to three, a a b, b a a, a b a, which may be expressed by 1x2x3. We have also

1 x 2

seen that the letters a b c d admit of 24 variations; but if we have a a b b, the 24 variations are reduced to six, viz. a a bb, abba, a bab, bba a, ba a b, baba, and this number of variations may be expressed by Hence, we have, as follows,

PROBLEM II. To find the number of changes that may be made in the arrangement of a given number of things, whereof there are several things of one sort, several of another, &c.

RULE. Take the natural series of numbers from 1 up to the given number of things, as if they were all different, and find the product of the terms.

Then take the natural series from 1 up to the number of similar things of one sort, and the same series up to the number of similar things of a second sort, &c., and divide the first product by the joint product of all these series, and the quotient will be the answer.

8. Find how many changes can be made in the order of the letters a a a b b c.

If the letters in this question were all different, they would admit of 1×2×3×4×5×6=720 variations; but since a is found 3 times, we must divide that number of variations by 1×2×3; and, since b occurs twice, we must again divide by 1X2; therefore the number of variations will be 1x2x3 X4 X5 X6. = 60.

1X2 X3 XIX 2

9. How many changes can be made in the order of the letters a aabbbbccdee?

10. How many variations may take place in the succession of the following musical notes, fa, fa, fa, sol, sol, la, mi, fa?

11. How many whole numbers can you make out of the number 1220055055, using all the figures each time? 12. How many variations can be made in the order of the figures in the number 97298279289 ?

PROBLEM III. Any number of different things being given, to find how many changes can be made out of them, by taking a given number of the things at a time.

RULE. Take a series of numbers commencing with the given number of things and decreasing by 1, till the number of terms is equal to the number of things to be taken at a time, and the product of all the terms of this series will be the answer.

To illustrate the rule, we will take the four letters abcd, and find the number of variations that can be

made upon them, by taking two at a time. In the first place, we will write the letter a on the left hand of each of the other letters, and the variations will be three, viz. a b, a c, a d; we will do the same with each of the other letters, thus, b a, b c, b d; c a, c b, c d; d a, d b, d c. Now we have all the changes that can be made upon the four letters, taking two at a time, and they are 4X3=12. We will also find, in the same manner, how many changes can be made on the same four letters, by taking three at a time; writing a on the left, thus, a b c, a b d; a cb, a cd; ad b, ad c, we have 3×2 6 variations. Now, since each of the letters is to be written in the same manner on the left, we shall have four such classes of variations, and the whole number will be 4X3X2=24 variations.

13. How many changes can be made upon the letters abcdef, by taking three at a time?

14. How many different whole numbers can be expressed by the nine digits, by using two at a time?

15. How many different whole numbers can be expressed by the nine digits, by using four at a time?

16. How many different numbers can you express with the nine digits and a cipher, by using five at a time?

XXXVII.

COMBINATIONS.

COMBINATION consists in taking a less number of things out of a greater without any regard to the order in which they stand. This is sometimes called Election or Choice.

No two combinations can have the same quantities; for instance, the quantities, a and b, admit of only one combination, because a b and b a are composed of the same quantities; but, if a third quantity c be added, we can make three combinations of two quantities out of them, because the third quantity c may be added to each of the two former, thus, a b, a c, b c; this number of combinations may be expressed by ix. If we add a fourth letter,

3X2

1X2'