A Course of Mathematics

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

Ans. 276.

3. A general, who had often been successful in war, was asked by his king what reward he should confer upon him for his services; the general only desired a farthing for every file, of 10 men in a file, which he could make with a body of 100 men; what is the amount in pounds sterling?

Ans. 18031572350/ 98 2d.

Then it is plain, that when m = 3, or the things to be combined are a,b,c, there can be only one combination. But if m be increased by 1,or the things to be combined are 4, as a, b, c, d, then will the number of combinations be increased by 3: since 3 is the number of combinations of 2 in all the preceding letters, a, b, c, and with each two of these the new letter d may be combined.

The number of combinations, therefore in this case, is 1 + 3. Again, if m be increased by one more, or the number of letters be supposed 5; then the former number of combinations will be increased by 6, that is, by all the combinations of 2 in the 4 preceding b, c, d; since, as before, with each two of these the new letter may be combined.

letters, a,

The number of combinations, therefore, in this case, is 1 + 3 + 6. Whence, universally, the number of combinations of m things, taken 3 by 3, is 1+ 3 + 6 + 10 &c. to m - 2 terms.

But the sum of this series is

m—1 m- 2

the same as the rule.

And the same thing will hold, let the number of things to be taken at a time be what it will; therefore the number of combinations of m things, taken n at a time, will be

m m-1 m-2 m

-X--X -X.

1 2

&c. to n terms. Q. E. D.

PROE

RROBLEM VI.

To find the Number of Combinations of any Given Number of Things, by taking any Given Number at a time; in which there are several Things of one Sort, several of another, &c.

RULE.

FIND, by trial, the number of different forms which the things to be taken at a time will admit of, and the number of combinations there are in each.

Add all the combinations, thus found together, and the sum will be the number required.

EXAMPLES.

1. Let the things proposed be a a abbe; it is required to find the number of combinations made of every 3 of these quantities?

2. Let a a ab bbc c be proposed; it is required to find the number of combinations of these quantities, taken 4 at a time? Ans. 10.

3. How many combinations are there in a a aa b b c c d e, taking 8 at a time? Ans. 13.

4. How many combinations are there in a aa a a b b b b b ccccddddee e efffg, taking 10 at a time? Ans. 2819.

PROBLEM. VII.

To find the Compositions of any Number, in an equal Number* of Sets, the Things themselves being all different.

RULE*.

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

* Demonstr. Suppose there are only two sets; then, it is plain, that every quantity of the one set being combined with every quantity of the other, will make all the compositions, of two things in these two sets

and

EXAMPLES.

1. Suppose there are four companies, in each of which there are 9 men; it is required to find how many ways 9 men may be chosen, one out of each company?

729

6561 the Answer.

Or, 9 × 9 × 9 × 9 6561 the Answer.

2. Suppose there are 4 companies; in one of which there are 6 men, in another 8, and in each of the other two 9; what are the choices, by a composition of 4 men, one out of each company?

Ans. 3888.

3. How many changes are there in throwing 5 dice?

Ans. 7776.

and the number of these compositions is evidently the product of the number of quantities in one set by that in the other.

Again, suppose there are three sets; then the composition of two, in any two of the sets, being combined with every quantity of the third, will make all the compositions of three in the three sets. That is, the compositions of two in any two of the sets, being multiplied by the number of quantities in the remaining set, will produce the compositions of three in the three sets; which is evidently the continual product of all the three numbers in the three sets.

And the same manner of reasoning will hold, let the number of sets be what it will. Q. E. D.

The doctrine of permutations, combinations, &c. is of very extensive use in different parts of the Mathematics; particularly in the calculation of annuities and chances. The subject might have been pursued to a much greater length; but what is here done, will be found sufficient for most of the purposes to which things of this nature are appli

cable.'

PRACTICAL

PRACTICAL QUESTIONS IN ARITHMETIC.

QUEST. 1. The swiftest velocity of a cannon-ball, is about 2000 fect in a second of time. Then in what time, at that rate, would such a ball be in moving from the earth to the sun, admitting the distance to be 100 millions of miles, and the year to contain 365 days 6 hours.

Ans. 8.4808

13149 years. QUEST. 2. What is the ratio of the velocity of light to that of a cannon-ball, which issues from the gun with a velocity of 1500 feet per second; light passing from the sun to the earth in 7 minutes? Ans. the ratio of 7822223 to 1.

QUEST. 3. The slow or parade-step being 70 paces per minute, at 28 inches each pace, it is required to determine at what rate per hour that movement is? Ans. 11 miles.

QUEST. 4. The quick-time or step, in marching, being 2 paces per second, or 120 per minute, at 28 inches each; then at what rate per hour does a troop march on a route, and how long will they be in arriving at a garrison 20 miles distant, allowing a halt of one hour by the way to refresh? the rate is 3 miles an hour. and the time 74 hr. or 7 h 17 min.

Ans.

QUEST. 5. A wall was to be built 700 yards long in 29 days. Now, after 12 men had been employed on it for 11 days, it was found that they had completed only 220 yards of the wall. It is required then to determine how many men must be added to the former, that the whole number of them may just finish the wall in the time proposed, at the same rate of working? Ans. 4 men to be added.

QUEST. 6 To determine how far 500 millions of gui neas will reach, when laid down in a straight line touching ⚫ one another; supposing each guinea to be an inch in diameter, as it is very nearly. Ans. 7891 miles, 728 yds, 2ft, 8 in.

QUEST. 7. Two persons, A and B, being on opposite sides of a wood, which is 536 yards about, they begin to go round it, both the same way, at the same instant of time; A goes at the rate of 11 yards per minute, and в 34 yards in 3 minutes; the question is, how many times will the wood be gone round before the quicker overtake the slower ?

Ans. 17 times.

QUEST.

ALGEBRA.

DEFINITIONS AND NOTATION.

ALGEBRA

LGEBRA is the science of computing by symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.

2. In this science, quantities of all kinds are represented by the letters of the alphabet. And the operations to be performed with them, as addition or subtraction, &c, are denoted by certain simple characters, instead of being expressed by words at length.

3. In algebraical questions, some quantities are known or given, viz. those whose values are known: and others unknown, or are to be found out, viz. those whose values are not known. The former of these arc represented by the leading letters of the alphabet, a, b, c, d, &c; and the latter, or unknown quantities, by the final letters, z, y, x, ù, &c.

4. The characters used to denote the operations, are chiefly the following:

+ signifies addition, and is named plus.

signifies subtraction, and is named minus.

X or signifies multiplication, and is named into.