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1 X2 X3 X 4 X 5 X 6 (= the number to be taken at a time )
10 X 9 X 8 X 7 X 6 X 5 ( =same number from 10)
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. 180315723501 98 2d,
Then it is plain, that when m =
3, or the things to be combined are 4, 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 to 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 letters, a, b, c, d ; since, as before, with each two of these the new letter c may be combined.
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.
2 But the sum of this series is х
X; which is 1 2
3 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 ni things, taken n at a time, will be =
m - 3 -X
&c. to n terms. & E. D. 1
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.
· 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.
1. Let the things proposed be a a a bbc; it is required to find the number of combinations made of every 3 of these quantities? Forms.
1 @?b, ac, b*a, 620
2. Let a a a bbb cc 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 bbcode, taking 8 at a time?
Ans. 13. 4. How many combinations are there in aaaaabbbbb ccccddddeee efff g, taking 10 at a time? Ans. 2819
To find the Compositions of any Number, in an equal Number
of Sets, the Things themselves being all different.
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, wul make all the compositions, of two things in these two sets”;
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?
6561 the Answer.
Or, 9 X 9 X 9 X 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?
and the number of these compositions is evidently the product of the pumber 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 applicable.
PRACTICAL QUESTIONS IN ARITHMETIC.
Quest. 1. The swiftest velocity of a cannon-ball, is about 2000 feet in a second of time. Then in what time, at that Tate, 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. 84808 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 75 minutes ?
Ans. the ratio of 782222 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. 113 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 garison 20 miles distant, allowing a halt of one hour by the way to refresh?
Sthe rate is 31 miles an hour.
and the time 7 hr. or 7 h. 174 min. 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 being to go round it, both the same way, at the same instant of time ; A goes at the rate of Il yards per minute, and B 34 yards in 3 minutes; and the question is, how many times will the wood be gonc round before the quicker overtake the slower?
Ans, 17 times.
Quest. 8. A can do a piece of work alone in 12 days, and B alone in 14 ; in what time will they both together perform a like quantity of work ?
Ans. 6 Is days. Quest. 9. A person who was possessed of a share of a copper mine, sold of his interest in it for 18001; what was the reputed value of the whole at the same rate ? Ans. 40001.
Quest. 10. A person after spending 201 more than 1 of his yearly income, had then remaining 301 more than the half of it; what was his income?
Ans. 2001. Quest. 11. The hour and minute hand of a clock exactly together at 12 o'clock; when are they next together?
Ans at 1 hr or 1 hr, is min. Quest. 12. If a gentleman whose annual income is 15001, spends 20 guineas a week; whether will he save or run in debt, and how much in the year?
Ans. save 4081. Quest. 13. A person bought 180 oranges at 2 a penny, and 180 more at 3 a penny ; after which, selling them out again at 5 for 2 pence, whether did he gain or lose by the bargain?
Ans. he lost 6 pence. Quest. 14. If a quantity of provisions serves 1500 men 12 weeks, at the rate of 20 ounces a day for each man ; how many men will the same provisions maintain for 20 weeks, at the rate of 8 ounces a day for each man? Ans. 2250 men.
Quest. 15. In the latitude of London, the distance round the earth, measured on the parallel of latitude, is about 15550 miles ; now as the earth turns round in 23 hours 56 minutes, at what rate per hour is the city of London carried by this motion from west to east? Ans. 64936, miles an hour.
Quest. 16. A father left his son a fortune, 1 of which he ran through in 8 months ; of the remainder lasted him 12 months longer ; after which he had bare 8201 left. What sum did the father bequeath his son ? Ans. 19131 68 8d.
Quest. 17. If 1000 men, besieged in a town with provisions for 5 weeks, allowing each man 16 ounces a day, be reinforced with 500 men more; and supposing that they cannot be relieved till the end of 8 weeks, how many ounces a day must each man have, that the provision may last that time?
Ans. 6 ounces. Quest. 18. A younger brother received 84001, which was just of his elder brother's fortune : What was the father worth at his death?