PERMUTATIONS AN D

TIONS.

COMBYA

The Permutation of Quantities is the shewing how many different ways 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 fame fituation.

The Combination of Quantities is the thewing how often a lefs number of things can be taken out of a great er, and combined together, without confidering their places, or the order they stand in.

This is fometimes called election or choice; and here every parcel must be different from all the reft, and ne two are to have precifely the fame quantitics or things.

The Compofition of Quantities is the taking of a given number of quantities out of as many equal rows of different quantities, one out of every 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 of things.

PROBLEM 1.

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 feries of numbers, from I up to the given number, continually together and the last product will be the anfwer required, EXAMPLES.

* Any two things a and b are capable of two variations only; as ab, ba; whose number is exprodled by 1.2.

If there be three things, a. b and, then any two of them, leaving out the third, will have 1X2 variations; and confequently when the third is taken in, there will be IX-Xj va riations, and fo on, as far as you please.

1.

EXAMPLES.

Christ Church, in Boston, has 8 bells; how many changes may be rung on them?

2.

1×2×3×4×5×6×7×8=40320 the Anf.

Nine gentlemen met at an inn, and were fo pleafed with their host, and with each other, that in a frolick they agreed to tarry so long as they, together with their host, could fit every day in a different pofition at dinner; pray how long, had they kept their agreement, would their frolick have lasted?

335

Anf 9941 years. 3. How many changes or variations will the alphabet admit of? Ans. 620448401733239439360000.

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

RULE. Take a series of numbers, beginning at the number of things given, and decreasing by one, to the number of quantities to be taken at a time; the product of all the terms will be the anfwer required.

EXAMPLES.

1. How many changes may be rung with 4 bel's out of 8 ?

2.

8×7×6×5(=4terms)=1680 the Anf.

How many words can be made with 6 letters of the alphabet, admitting a number of confonants may make a word ? Anf. 96909120.

Any number of things being given whereof there are feu-
eral things of one fort, feveral of another, &c. To find
how many changes may be made out of them all.
RULE*

1. "Take the series 1X2X3X4, &c. up to

the

the number of things given, and find the product of the terms.

2. Take the series 1X2X3X4, &c. up to the numbe of the given things of the first sort, and the feries, 1X2X3X4, &c. up to the number of the given things of the second fort, &c.

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

EXAMPLES.

I. How many variations may be made of the letters

in the word Zaphnathpaaneah?

IX2x3X4X5x6x7x8X9X10X11X12X13×14×15(=number

of letters in the word)

1307674368000.

1X2X3X4X5(=number of a's)=120

1X2(number of p's)=

2

I

1 (number of t's)=== X2X3(number of h's) = 6 1X2(=number of n's)=

2

2X6x1x21x20=2880) 1307674368000(454053600-the Ans. How many different numbers can be made of

2.

of the

following figures, 1223334444? Ans. 12600.

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 series 1, 2, 3, 4, &c. up to

the

Any 3 quantities, a, b, c, all different from each other, admit of 6 variations; but if the quantities are all alike, or abc become ana then the 6 variations will be reduced to 1, which may be expressed by Ix2x3 I. Again if two quantities out of three are alike,

1xIX3

or, abc, become aac; then the 6 variations will be reduced to these 3, aac, caa, aca, which may be expressed by 1x2x3=3, and fo

of any greater number.

IX2

In any given number of quantities, the number of combina

ncreases gradually until you come about the mean num

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

2.

Take a feries of as many terms, decreasing 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 last product by the former, and the quotient will be the number fought.

EXAMPLES.

1. How many combinations may be made of 7 let

ters out of 12?

1X2X3X4X5X6X7 (=the num

ber 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 asked by his king, what reward he should confer on him for his services; 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 75. IITI

65

4. A farmer bargained with a gentleman for a dozen sheep (at 2 dollars per head) which were to be picked out of 2 dozen; but being long in choosing 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 6s. 4d. PROBLEM

bers, and then gradually decreases. If the number of quantities be even, half the number of places will thew the greatest number of combinations, that can be made of those quantities; but if odd, then those two numbers, which are the middle, and whose sum - is equal to the given number of quantities, will thew 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 ?

9X9X9×9×9=59049 different ways. 2. How many chances are there in throwing 4 dice ? As a die has 6 fides, multiply 6 into itself 3 times continually. 6X6X6X6=1296 chances, Anf.

3. Suppose a man undertakes to throw an ace at one throw with 4 dice: what is the probability of his effecting it?

First, 6X6x6X6=1296 different ways with and without the ace ;

Then, if we exclude the ace side of the die, there will be five sides left, and 5X5X5X5=625 ways without the ace; therefore there are 1296-625=671 ways, wherein one or more of them may turn up an ace: and the probability that he will do it, as 671 to 625, Anf.

THE USE OF LOGARITHMS.

1. IN MULTIPLICATION.

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

RULE. To the logarithm of 275, viz.

product.

Add the logarithm, of 12,6, viz.

And their fum is the logarithm

2,43933

1,10037

3465-3,53970

2. IN DIVISION.

Let it be required to find the quotient, which arifes by

dividing one number by another; suppose 1425 by 57

From