the number of things given, and find the product of all the terms. 2. Take the feries 1X2 X3X4, &c. up to the number of the given things of the first fort, and the feries, 1X2X3X4, &c. up to the number of the given things of the fecond fort, &c. 3. Divide the product of all the terms of the first feries by the joint product of all the terms of the remaining ones and the quotient will be the answer required. EXAMPLES. 1. How many variations may be made of the letters in the word Zaphnathpaaneah? 1X2×3X4X5×6×7×8X9X10X11 × 12X13× 14 × 15(=number of letters in the word) = =1307674368000. 2 I 6 2 Anf. Anf, 12600. 2. How many different numbers can be made of the following figures, 1223334444 ? PROBLEM 4. 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 expreffed by IX2X3 1x1x3 I. Again if two quantities out of three are alike, or, abc, become aac; then the 6 variations will be reduced to thefe Ix2x3 3, aac, caa, aca, which may be expressed by- =3, and fo of any greater number. IX2 In any given number of quantities,the number of combinations increafes gradually until you come about the mean num bers, the number to be taken at a time, and find the product of all the terms. 2. Take a series 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 laft product by the former, and the quotient will be the number fought. EXAMPLES. 1. How many combinations may be made of 7 letters out of 12? 1X2X3X4X5X6X7 (=the number to be taken at a time,) 12 X 11 X10X9X8X7X6 (=same 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 afked by his king, what reward he fhould confer on him for his fervices; 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 7811T124 65 4. A farmer bargained with a gentleman for a dozen. fheep (at 2 dollars per head) which were to be picked out of 2 dozen; but being long in choofing 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 fhould have the whole, to which the farmer readily agreed: Pray what did they coft him? Anf. 11267 6s. 4d. PROBLEM bers, and then gradually decreases. If the number of quanti ties be even, half the number of places will fhew the greatest number of combinations, that can be made of thofe quantities; but if odd, then those two numbers, which are the middle, and whofe fum is equal to the given number of quantities, will hew the greatest number of combinations. FF 2 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 fet continually together, and the product will be the answer required. EXAMPLES. 1. Suppose there are 5 companies, each consisting of 12 men it is required to find how many ways 9 men be chofen one out of each company ? may 9X9X9X9X9=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. Suppofe 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 fide of the die, there will be five fides 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 product. RULE. To the logarithm of 275, viz. Add the logarithm, of 12,6, viz. And their fum is the logarithm} of their product, viz. 2. IN DIVISION. 2,43933 1,10037 3465=3,53970 Let it be required to find the quotient, which arifes by dividing one number by another; fuppofe 1425 by 57. From = Fromthe logarithm of the dividend, viz. 14253,15381 } 3. IN THE RULE OF THREE. 25 1,3979 Three numbers given to find a fourth, in direct proportion. RULE. From the tables take the logarithms of each of the propofed numbers, then add the logarithms of the fecond and third together, and from the fum take the logarithm of the first and the remainder will be the logarithm of the fourth number. Let the three propofed numbers be 18, 24 and 33, and the operation will stand thus: 1,3802f the logarithm of 24, the 2d. term. 1,51851the logarithm of 33, the 3d. term. 2,89872 the logarithm of their product. -1,25527 the logarithm of the ift term 18. 1,64345 the logarithm of the 4th term required, which, by the Table anfwers to the natural number 44, the 4th proportional to the three propofed numbers. 4. IN INVOLUTION, OR RAISING POWERS. To find any power of any proposed number, or to involve any number to any propofed power, by logarithms. RULE. Multiply the logarithm of the given root by the power, viz. by 2 for the fquare, by 3 for the cube,&c. and the product is the logarithm of the power fought. Required to find the cube of 12 ? 1,07918 the logarithm of 12. X3 the third powers or cube. 3,23754 1728 the cube of 12. 5. IN EVOLUTION, OR EXTRACTING ROOTS. To Extract any root of any proposed number. RULE. Divide the logarithm of the propofed number by the index of the required root, viz. by 2 for the fquare, by 3 for the cube, &c. and the quotient will be the logarithm of the root required. “ Required Required to find the cube root of 1728 ? 3,23754 the logarithm of 1728, and 3,23754÷ 31,07918 is the logarithm of the cube root of 1728, viz. 12. 6. IN COMPOUND INTEREST. To find the amount of any fum for any time, and at any rate at compound intereft. RULE. Multiply the logarithm of the ratio (i. e. the amount of 11. for 1 year) by the number of years, and to the product add the logarithm of the principal; the fum will be the logarithm of the amount. What will 451. amount to forborne 12 years, at 6 per cent. per annum, compound interest ? Log. of 1,06, the ratio, is 02533 Multiply by the time 12 ,30396 Add Log. of 45, the principal 1,65321 7. The fum is 1,95717 which is the logarithm of 90,7=£90 14s. Auf. IN DISCOUNT AT COMPOUND INTEREST. To find the prefent worth of any fum of money, due any time hence, at any rate, at compound intereft. RULE. From the logarithm of the fum to be dif counted, fubtract the logarithm of the rate multiplied by the time; and the remainder is the logarithm of the prefent worth. What prefent money will pay a debt of 90l. 14s. due 12 years hence, difcounting at the rate of 6 per annum ? From the logarithm of 9ol. 141,95717 Sub. product of the Log. of the ratio X by the time. } = 30396 per cent. The remainder 1,65421 is the logarithm of 451. Anf. N. Ham. |