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the number of things given, and find the product of al

the terms.

2. Take the feries 1 X2 X3 X4, &c. up to the number of the given things of the first fort, and the feries, 1X2 X3 X4, &c. up to the number of the given things of the fecond fort, &c.

3. Divide the product of all the terms of the first feries by the joint product of all the terms of the remaining ones, and the quotient will be the answer required.

E x A MPLE S.

1. How many variations

may be made of the letters

in the word Ziphnathpaanesh ?

X2X3X4X5X6X7X8X9X10X11X12X13X14X15

aumber of letters in the word)=1307674368000. 1X2X3X4X5 (=number of a's) 120 1X2 (=number of p's)=

I

(number of t's)=

2

1X1X3 (=number of h's) 6 1×2 (=number of n's) = 2 2X6X1X21X20=2880)1307674368000(454053600-the

2. How many different numbers can be made of the following figures, 1223334444?

PROBLEM

Anf.

Anf. 12606.

4.

To find the number of combinations of any given number of things, all different from one another, taking`one given number at a time.

RULE * I -Take the feries 1, 2, 3, 4, &c. up to

the

Any 3 quantities, a, b, c, all different from each other, admit of & variations; but if the quantities are all alike, or be became aaa, then the 6 variations will be reduced to 1, which may be expressed . Again, if two quantities out of three are alike, or, abc, become aur; then the 6 variations will be reduced to thefe

by

1X2X3

IX2X3

3, aar, caa, pca, which may be expreffed by of any greater number.

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In any given number of quantities, the number of combinations increases gradually until you come about the mean numbers,

and

the number to be taken at a time, and find the product of all the terms.

2. Take a series of as many terms, decreafing by 1, from the given number, out of which the election is to be made, and find the product of all the terms.

3. Divide the laft product by the former, and the quotient will be the number fought.

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EXAMPLES.

1. How many combinations may be made of 7 letters out of 12?

1X2 X3 X4 X5×6×7 (the number to be taken at a time,) 12X11X10X9x8x7x6 (fame number from 12.)

5040

=3991680.

5040)3991680(792 the Anf.

2. How many combinations can be made of 6 letters out of the 24 letters of the alphabet?

Anf. 134596.

3. A general was afked by his king, what reward he fhould confer on him for his fervices; the general only required a penny for every file, of 10 men in a file, which he could make out of a company of 90 men: What did it amount to? Anf. £23836022841 7/11

65

4. A farmer bargained with a gentleman for a dozen fheep (at 2 dollars per head) which were to be picked out of a dozen; but being long in choofing them, the gentleman told him, that if he would give him a penny for every different dozen which might be chofen out of the two dozen, he should have the whole, to which the farmer readily agreed: Pray what did they cost him? Anf. £11267 6f4. PROBLEM

and then gradually decreafes. If the number of quantities.be even, half the number of places will fhew the greatest number of combinations, that can be made of thofe quantities; but if odd, then thofe two numbers, which are the middle, and whofe fum is equal to the given number of quantities, will fhew the greatest number of combinations.

PROBLEM 5.

To find the compofitions of any number, in an equal number of fets, the things being all different.

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

EXAMPLES...

1. Suppose there are 5 companies, each confifting of 12 men: It is required to find how many ways 9 men may be chofen, one out of each company?

9×9×9×9×9=59049

differert ways. 2. How many chances are there in throwing 4 dice? As a die has 6 fides, multiply 6 into itself three times continually. 6×6×6×6=1296 chances, Anf. 3. Suppofe a man undertakes to throw an ace at one throw with 4 dice; what is the probability of his effecting it?

Firft, 6x6x6x6=1296 different ways with and without the ace ;

Then, if we exclude the ace fide of the die, there will He five fides left and 5 × 5 × 5 × 5625 ways without the ace; therefore there are 1296-625-671 ways, wherein one or more of them may turn up an acc: And the probability that he will do it, as 671 to 625, Anf..

THE USE OF LOGARITHMS.

I. IN MULTIPLICATION.

Given two numbers, viz. 275 and 12,6 to find their

product.

RULE. To the logarithm of 275, viz.

Add the logarithm, of 12,6, viz.

And their fum is the logarithm

of their product, viz.

2. IN DIVISION.

2,43933

1,10037

3465=3,53970

Let it be required to find the quotient, which arises by, dividing one number by another; fuppofe 1425 by 57.

From

From the logarithm of the dividend, viz. 1425=3,15381
Take the logarithm of the divifor, viz.

And the remainder
rithm of the quotient, viz.

57 1,75587

3. In the RULE of THREE.

25=1,39794

Three numbers given to find a fourth, in direct proportion. RULE. From the Tables take the logarithms of each of the propofed numbers, then, add the logarithms of the fecond and third together, and from the fum take the logarithm of the first, and the remainder will be the logarithm of the fourth number.

Let the three propofed numbers be 18, 24 and 33, and the operation will stand thus ;

1,38021 the logarithm of 24, the 2d term.
1,51851 the logarithm of 33, the 3d term.
2,89872 the logarithm of their product.
1,25527 the logarithm of the first term 18.

1,64345 the logarithm of the 4th term required, which, by the Table, anfwers to the natural number 44, the 4th proportional to the three propofed numbers.

4. In INVOLUTION or RAISING POWERS.

To find any power of any propofed number, or to involve any number to any propojed power, by logarithms.

RULE.-Multiply the logarithm of the given root, by the power, viz. by 2 for the fquare, by 3 for the cube, &c. and the product is the logarithm of the power fought. Required to find the cube of 12?

1,07918 the logarithm of 12..

X3 the third power, or cube.

3,23754 1728, the cube of 12.

5. In EVOLUTION, or EXTRACTING ROOTS.

To Extract any Root of any propofed number.

RULE. Divide the logarithm of the propofed number by the index of the required root, viz. by 2 for the fquare, by 3 for the cube, &c. and the quotient, will be the log, arithm of the root required.

Required

Required to find the cube root of 1728?

3,23754 the logarithm of 1728, and 3,23754÷ 3=1,07918 is the logarithm of the cube root of 1728,

viz. 12.

6. In COMPOUND INTEREST.

To find the amount of any fum for any time, and at any rate, at compound intereft.

RULE. Multiply the logarithm of the ratio (i. e. the amount of 1/. for 1 year) by the number of years, and to the product add the logarithm of the principal; the fum will be the logarithm of the amount.

What will 45%. amount to, forborne 12 years, at 6 per cent. per annum, compound interest ?

Log. of 1,06, the ratio, is ,02533
Multiply by the time

12.

30396

Add log. of 45, the principal 1,65321

The fum is 1,95717 which is the logarithm of 90,7=£90 14s. Anf.

7. In DISCOUNT, at COMPOUND INTEREST. To find the prefent worth of any fum of money, due any time bence, at any rate, at compound intereft.

RULE. From the logarithm of the fum to be dif counted, fubtract the logarithm of the rate multiplied by the time; and the remainder the logarithm of the pref

ent worth.

What present money will pay a debt of gol. 145. due 12 years hence, discounting at the rate of 6 per cent, per annum ?

From the logarithm of £90 141,95717 Sub. product of the Log. of the ratio by the time

•}=

=,30396.

The remainder 1,65421 is the logarithm of £45 Ans. N. Ham.

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