PROBLEM 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 aabbe; it is required to find the number of combinations made of every 3 of these quantities? Forms. аз a3b, a2c, ba, b2c a b c Combinations. 1 4 1 Number of combinations required =6 - 2. Let a aabbb c 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 a a b b c c d e, taking 8 at a time? Ans. 13. 4. How many combinations are there in a a a 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 the number of these compositions EXAMPLES. 1. Suppose there are four companies, in each of which there are 9 men; it is required to find how many ways 4 men may be chosen, one out of each company? 9 9 81 9 729 9 6561 the Answer. Or, 9X9X9x9-6561 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. 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. g. 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 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 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. 84 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? Ans. { the rate is 3,2 miles an hour. and the time 72 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 guineas 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 yards. 2 ft. 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 11 yards per minute, and в 34 yards in 3 minutes; and the question is, how many times will the wood be gone round before the quicker overtake the slower? Ans. 17 times. QUEST. QUEST. 8. A can do a piece of work alone in 12 days, and alone in 14; in what time will they both together perform a like quantity of work? Ans. 6 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. 4000l. QUEST. 10. A person after spending 201 more than 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 are exactly together at 12 o'clock; when are they next together? Ans. 1hr, or 1 hr 5 min. QUEST. 12. If a gentleman whose annual income is 15007, 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 mantain 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. 649 33 miles an hour. 359 QUEST. 16. A father left his son a fortune, 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. 1913/ 6s 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. 63 ounces. QUEST. 18. A younger brother received 8400/, which was just of his elder brother's fortune: What was the father worth at his death? Ans. 192001. QUEST. QUEST. 19. A person, looking on his watch, was asked what was the time of the day, who answered, It is between 5 and 6; but a more particular answer being required, he said that the hour and minute hands were then exactly together; What was the time? Ans. 27 min. past 5. QUEST. 20. If 20 men can perform a piece of work in 12 days, how many men will accomplish another thrice as large in one-fifth of the time? Ans. 300. QUEST. 21. A father devised of his estate to one of his sons, and of the residue to another, and the surplus to his relict for life. The children's legacies were found to be 514/ 6s 8d different: Then what money did he leave the widow the use of? Ans. 1270 1s 91дd. QUEST. 22. A person, making his will, gave to one child 13 of his estate, and the rest to another. When these legacies came to be paid the one turned out 1200l more than the other : What did the testator die worth? Ans. 4000l. QUEST. 23. Two persons, A and B, travel between London and Lincoln, distant 100 miles, A from London, and в from Lincoln, at the same instant. After 7 hours they meet on the road when it appeared that a had rode 11 miles an hour more than B. At what rate per hour then did each of the travellers ride? Ans. A 73, and 6 miles. QUEST. 24. Two persons, A and B, travel between London and Exeter. A leaves Exeter at 8 o'clock in the morning, and walks at the rate of 3 miles an hour, without intermission and B sets out from London at 4 o'clock the same evening, and walks for Exeter at the rate of 4 miles an hour constantly. Now, supposing the distance between the two eities to be 130 miles, whereabouts on the road will they meet? Ans. 693 miles from Exeter. QUEST. 25. One hundred eggs being placed on the ground. in a straight line, at the distance of a yard from each other: How far will a person travel who shall bring them one by one to a basket, which is placed at one yard from the first egg? Ans. 10100 yards, or 5 miles and 1300 yds. QUEST. 26. The clocks of Italy go on to 24 hours; Then how many strokes do they strike in one complete revolution of the index? Ans. 300. QUEST. 27. One Sessa, an Indian, having invented the game of chess, showed it to his prince, who was so delighted |