RÚLE. Multiply all the terms, from one up to the givea number, continually together, and the last product will be the number of changes. EXAMPLES. { 1. Suppose three men in a room, Tom, Jack and Bill; to how many different positions can you place them ? 1st position, Tom, Jack and Bill. Bill, Tom and Jack. 1 X%XS 6 Ans. 2. How many changes may be rung on 12 bells, and also kow long they would be in ringing but once over, supposing 24 changes might be rung in a minute, and the year to consist of 365 days 6 hours 479001600 changes. Ans. 37 years 49 weeks 2 days 18 hours. 3. Seven gentlemen, who were travelling, met together by chance at a certain inn upon the road, where they were so well pleased with their host, and each other's company, that in a frolic they offered him 150 dollars, to stay at that place so long as they, together with himself, could sit every day at dinner in a different order or position. The host, thinking that they could not sit in many different positions, because there were but a few of them, and that himself could make no considerable alteration, he being but one, imagined that he should make a good bargain, and readily, for the sake of a good dinner, and better company, entered into an agreement with them, and so made himself the eighth per I demand how long they staid at that inn, and how many different positions they sat in? Ans. They staid till they buried their host and his son, and the time was 110 years 142 days, allowing 365 days 6 hours to the year. The number of positions were 40320. What number of changes will the nine digits allow of? 1x2x3x4x5x6x7x8x9=362880 Ans. 5. In how many different positions can you place the letters that spell Washington ? Ans. 3628800. son. COMBINATION. Combination of numbers or things, is the shewing how often a less number of things may or can be taken out of a greater, and combined or joined together differently. RULE. Take 1, 2, 3, 4, &c. up to the number given to be combined; take another rotation downwards, or decreas. ing by unity, from the number out of which the combinations are to be made. Multiply the first continually for a divisor, and the latter for a dividend, and the quotient will be the answer. EXAMPLES. 1. How many combinations can be made of 6 letters out of 10? Ans. 924. 1x2x3x4x5x6=720 divisor 12x11x10x9x8x7=665280 dividend 720)665280 Ans. 924 2. How many combinations may be made of 7 letters out of 122 Ans. 792. any, PROGRESSION ARITHMETICAL PROGRESSION Is rank or quantity of numbers, more than two, increasing by a common excess, or decreasing by a common difference; as 2, 4, 6, 8, &c. is called the ascending series in arithmetical progression, and 8, 6, 4, 2, &c. is called the descending series in arithmetical progression. A series in progression consists of five parts, viz. the first term, the last term, number of terms, common difference, and sum of the series. By having three of these parts given, the other two may be found, which admit of a variety of questions. The numbers which form the series, are termed or called the progressional terms; the first and last terms are called the extremes. In a series of even numbers, the sum of the two extremes will be equal to the sum of any two terms, equally distant from them; as 2, 4, 6, 8, 10, the two extremes being 2+10=12, so 4+8=12; but if the number of terms be odd, the double of the middle term will be equal to any two terms, equally distant therefrom, as 1, 3, 5, 7, 9, 11, 13. The double of the middle term, 7, equal 14, and 3+11=14, and 5+9=14. Progressions are best understood by Algebra. The first term, the last term, and the number of terms, giren to find the sum of all the terms. RULE. Multiply the sum of the two extremes by the number of terms, and half that product will be the answer; or multiply the sum of the two extremes by half the number of terms; or multiply half the sum of the two extremes by the whole number of terms, and the product will be the answer. The five things in progression are numbered as follows: 1. The first term. EXAMPLES. he 1. How many strokes does a clock strike in 24 hours? First for 12 hours. 124.1=13X6=78 the number for 12 hours, which multiplied by 2 equal 156 for 24 hours. 2. A man bought 17 yards of cloth, and gave for the first yard 40 cents, and for the last yard two dollars. What did pay for the cloth ? Ans, 20 dollars 40 cents. 3. If 100 apples were placed in a right line, exactly two yards asunder from one another, and the first two yards from a basket, what length of ground will that person travel, who gathers these apples singly, returning with every apple to the basket to put it in ? Ans. 11 miles 840 yards. 4. The first term of an arithmetical series is 4, the last 24, and number of terms 12. Required the sum of the series? 24+4=28 sum of the extremes; then 28 X 12-2=168 Ans. 5. A merchant bought 19 yards of muslin, and was to pay 20 cents for the first yard, and 7 dolls. 40 cts. for the last yard. What did the whole come to ? Ans. 72 dollars 20 cents, The first term, the last term, and the number of terms, given to find the common difference. RULE. Divide the difference of the extremes by the number of terms less one, and the quotient will be the common difference, or fourth term. Or, from the second term subtract the first, and the remainder divided by the third less one gives the fourth. 6. The extremes, (or first and second terms,) are 5 and 33, and the number of terms 15; what is the common difference, or fourth term ? 39-5“ 15-1=2 Ans. 7. A man had nine sons, whose several ages differ alike; the youngest was three years old, and the oldest thirty-five. What was the common difference of their ages ? Ans. 4 years. 8. A man is to travel from Washington to a certain place in 12 days, and to go but 3 miles the first day, increasing every day by an equal number of miles, so that his last day's journey inay be 58 miles. What is his daily increase, and the length of the whole journey? Ans. Daily increase 5 miles, and whole journey 366. 9. A debt is to be discharged at 16 different payments, in arithmetical progression; the first payment is to be 56, and the last 400 dollars. What is the common difference, and sum of the whole debt? Ans. 22 dolls. 93} cts. common difference. 2 8648 dollars, the whole debt. The first term, last term, and common difference, given to find the number of terms; or, which is the same, the first, second and fourth terms given to find the third. Rule. Divide the difference of the extremes by the common difference, add one to the quotient, and that sum is the number of terms; or, from the second subtract the first, the remainder divide by the fourth, and to the quotient add one, which gives the third. 10. If the extremes be 5 and 47, and the common difference 2, what is the number of terms ? 47-5+2+1 = 22 Ans. 11. A man, going a journey, travelled 5 miles the first day, 45 miles the last day, and each day increased his jour ney by 4 miles. How many days did he travel, and how far? Ans. 11 days, and 275 miles distạące, 12. A man being asked how many sons he had, said that the youngest was 6 years old, and the oldest 38, and that one was born every four years in his family. How many had he? 38—6=32;4+1=9 Ans. The last term, the number of terms, and the comraon differ ence, given to find the first term; or, the second, third and fourth to find the first term. Rule. Multiply the fourth by the third, made less by one, and the product subtracted from the second gives the first. 13. A man in 12 days went from Washington to a certain town in the country, every day's journey increasing the former by three miles, and the last day he went 48 miles. What was his first day's journey? 3x12–1533; then 48-33=15 miles, answer. 14. A man takes out of his purse, at eight several times, so many different numbers of dollars, every one exceeding the former by five, and the last was fifty. "What was the first? Ans. 15 dollars. The last term, the number of terms, and the sum of all the terms, given to find the first term; or, the second, third and fifth given to find the first term. RULE. Divide the fifth by the third, and from the quetient subtract half the product of the fourth, multiplied by the third less one, which gives the first. 15. A man is to receive 1440 dollars, at 12 several pay. ments, each payment to exceed the other by an equal excess of 16 dollars; what is the first payment? 14404-12= 120; then 16 X 12-1+2=88; then 120-88= 832 Ans. The first term, the number of terms, and the common differ ence, given to find the last term; or, the first, third and fourth given to find the second term. RULE. Subtract the fourth from the product of the third multiplied by the fourth, and that remainder added to the first gives the second. 16. What is the last number of an arithmetical progression, beginning at 5, and continuing, by an increase of 7, to 35 places ? 35x7=245-7=238+5=243 Ans, |