3. Two persons, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A's money. is now double that of B: what did each lay out?

Ans. 3001.

4. A laborer was hired for 40 days, on this condition, that he should receive 20d. for every day he wrought, and forfeit 10d. for every day he was idle; now he received at last 21. 1s. 8d.: how many days did he work, and how many was he idle?

Ans. He wrought 30 days, and was idle 10. 5. A gentleman has two horses of considerable value, and a saddle worth 501.; now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the first: what is the value of each horse? Ans. One 30l. and the other 401.

6. There is a fish, whose head is 9 inches long, and his tail is as long as his head and half as long as his body, and his body is as long as his tail and his head: what is the whole length of the fish? Ans. 6 feet.

PERMUTATION AND COMBINATION.

THE Permutation of Quantities is the showing how many different ways the order or position of any given number of things may be changed.

This is also called Variation, Alternation, or Changes; and the only thing to be regarded here is the order they stand in; for no two parcels are to have all their quantities placed in the same situation.

The Combination of Quantities is the showing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order they stand in.

This is sometimes called Election, or Choice; and here every parcel must be different from all the rest, and no two are to have precisely the same quantities, or things.

The Composition of Quantities is the taking a given number of quantities out of as many equal rows of different quantities, one out of each row, and combining them together.

Here no regard is had to their places; and it differs from combination only, as that admits of but one row, or set of things.

Combination of the same form are those, in which there is the same number of quantities, and the same repetitions: thus, abcc, bbad, deef, &c. are of the same form; but abbc, abbb, aacc, &c. are of different forms.

PROBLEM I.

To find the number of permutations, or changes, that can be made of any given number of things, all different from

each other.

RULE.*

Multiply all the terms of the natural series of numbers, from 1 up to the given number, continually together, and the last product will be the answer required.

EXAMPLES.

1. How many changes may be made with these three letters, abc ?

*The reason of the rule may he shown thus: any one thing a is capable only of one position, as a.

Any two things, a and b, are only capable of two variations; as ab, ba; whose number is expressed by 1×2.

If there be 3 things, a, b, and c, then any two of them, leaving out the third, will have 1×2 variations; aud consequently, when the third is taken in, there will be 1×2×3 variations.

In the same manner, when there are 4 things, every 3, leaving out the fourth, will have 1×2×3 variations. Then, the fourth being taken in, there will be 1×2×3×4 variations. And so on, as far as you please.

2. How many changes may be rung on 6 bells?

Ans. 720. 3. For how many days can 7 persons be placed in a different position at dinner? Ans. 5040 days.

4. How many changes may be rung on 12 bells, and how long would they be in ringing, supposing 10 changes to be rung in 1 minute, and the year to consist of 365 days 5 hours and 49 minutes?

Ans. 479001600 changes, and 91y. 26d. 22h. 41m. 5. How many changes may be made of the words in the following verse? Tot tibi sunt dotes, virgo, quot sydera cœlo. Tottibiswartes

PROBLEM II.

Ans: 40320.

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

RULE.*

Take a series of numbers, beginning at the number of things given, and decreasing by 1 till the number of terms

* This rule, expressed in terms, is as follows: mxm—1 X m-2xm-3, &c. to n terms; where m = number of things given, and n = quantities to be taken at a time.

be equal to the number of things to be taken at a time, and the product of all the terms will be the answer required.

In order to demonstrate the rule, it will be necessary to premise the following

LEMMA.

The number of changes of m things, taken n at a time, is equal to m changes of m-1 things, taken n-1 at a time.

DEMONSTRATION. Let any 5 quantities, abcde, be given.

First, leave out the a, and let v = number of all the variations of every two, bc, bd, &c. that can be taken out of the 4 remaining quantities, bcde.

Now let a be put in the first place of each of them, abc, abd, &c. and the number of changes will still remain the same; that is, vnumber of variations of every 3 out of the 5, abcde, when a is first.

In like manner, if b, c, d, e, be successively left out, the number of variations of all the twos will also =v; and b, c, d, e, being respectively put in the first place, to make 3 quantities out of 5, there will still be v variations as before.

But there are all the variations, that can happen of 3 things out of 5, when a, b, c, d, e, are successively put first; and therefore the sum of all these is the sum of all the changes of 3 things out of 5.

=

But the sum of these is so many times v, as is the number of things; that is, 5v, or mv, all the changes of three things out And the same way of reasoning may be applied to any numbers whatever.

DEMONSTRATION OF THE RULE. Let any 7 things, abcdefg, be given, and let 3 be the number of quantities to be taken. Then m7, and n=3.

Now it is evident, that the number of changes, that can be made by taking I by 1 out of 5 things, will be 5, which let =v. Then, by the lemma, when m=6 and n=2, the number of changes will =mv=6×5; which let v a second time.

Again by lemma, when m=7 and n=3, the number of changcs =mv=7x6x5; that is, mv=mxm-1x m-2, continued to 3, or n terms. And the same may be shown for any other num.

EXAMPLE.

1. How many changes may be made out of the 3 letters, abc, by taking 2 at a time.

2. How many words can be made with 5 letters of the alphabet, it being admitted, that a number of consonants may make a word? Ans. 5100480.

PROBLEM III.

Any number of things being given, whereof there are several given things of one sort, several of another, &c. to find how many changes can be made out of them all.

RULE.*

1. Take the series 1, 2, 3, 4, &c. up to the number of things given, and find the product of all the terms.

* This rule is expressed in terms thus :

1x2x3x4x5, &c. to m

1x2x3, &c. to ʼn x1×2×3, &c. to 7, &c.

; where m = number

of things given, p=number of things of the first sort, g = number of things of the second sort, &c.

The DEMONSTRATION may be shown as follows.

Any two quantities, a, b, both different, admit of 2 changes; but if the quantities be the same, or ab become aa, there will be

but one alteration, which may be expressed by.