2. the number of things given, and find the product of al the terms. Take the series 1 X2 X3 X4, &c. up to the name ber of the given things of the first fort, and the feries, 1X2 X 3 X 4, &c. up to the number of the given things of the second fort, &c. 3. Divide the product of all the terms of the first fe. ries by the joint product of all the terms of the remaining enes, and the quotient will be the answer required. Il20 llow may be made of the letters in the word Zaplanatbpaaneah ? iX2X3X 1X5X6X7X8X9X10X11X12X13X14X15 (= number of letters in the word)=1307674368000. 1X2X3X4X5 (=number of a's)= 1X2 (=number of p’s) number of t's). 1X2X3 (=number of h's) = 6 1X2(=number of n's)= 2 2X6X1X21 X20-2880)1 307674368000(454053600--thé Ans. How many different numbers can be made of the following figures, 1223334444? Anf. 12606. 2 I 2. 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 * -Take the series 1, 2, 3, 4, &c. up to the Any 3-quantities, a, b, c, all different from cach other, admit of 6 variations; but if the quantities are all alike, or br became aaa, then the 6 variations will be reduced to 1, which may be expresad Again, if two quantities out of three are alike, IX2X3 or, abc, become au?"; then the 6 variations will be reduced to thefe 1X2X3 3; nał, caa, roa, which may be expressed by of any greater number. iX2 ** In any given number of quantities, the number of combinations increases gradually until you come about the mean numbers, and by 'X2X3 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 lait product by the former, and the quotient will be the number sought. EX A M P L E S. 1. How many combinations may be made of 7 letters out of 12 ? IX2 X3 X4 X5 X6x7 (=the num 5040 ber to be taken at a time,) 12X11 X 10 X 9 X 8 X7 X6 (=fame number from 12.) =3991680. 5040) 3991680(792 the Ang. 2. How many combinations can be made of 6 letters out of the 24 letters of the alphabet ? Anf. 134596. 3. A general was asked by his king, what reward he should confer on him for his services; the general only required a penny for file, of 10 men in a file, which he could make out of a company of 90 men : What did it amount to? Anf: 623836022841 7/11 1111: 4. A farmer bargained with a gentleman for a dozen sheep (at 2 dollars per lead) which were to be picked out of a dozen ; but being long in choosing them, the gentleinan told him, that if he would give him a penny for every diferent dozen which might be chosen out of the two doze:1, he should have the whole, to which the farmer readily agreed : Pray what did they cest him? Anf £11267 654. and then gradually decreases. If the number of quantities.be even,' half the number of places will shew the greatest number of combinations, that can be made of those quantities; but if odd, then thnfe two numbers, which are the middle, and whose sun is equal to the given aumber of quantities, will shew the greatest number of combinations PROBLEM 5. To find the compositions of any number, in an equal number of fets, the things being all different. Rule.- Multiply the number of things in every set continually together, and the product will be the answer required. EXAMPLES. 1. Suppose there are 5 companies, each confisting of 12 men : It is required to find how many ways 9 men niay be chosen, one out of each.company ? 9X9X9X9x9=59049 differert ways, 2. How many chances are there in throwing 4 dice ? As a die has 6 fides, multiply 6 into itfelf three times continually. 6x6x6 X 6=1296 chances, Anf. 3. Suppofe a man undertakes to throw an ace at one throw with 4 dice ; what is the probability of his effect. ing it? First, 6x6x6x6=1296 different ways with and without the ace ; Then, if we exclude the ace side of the die, there will lie five fides left and 5X5X5X5=625 ways without the. ace; therefore there are 12964625=671 ways, where in one or more of them may turn up an ace : And the probability that he will do it, as 671 to 625, Ang. 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. 2,43933 Add the logarithm, of 12,6, viz. 1,10037 And their fuum is the logarithm of their product, viz. 3465=3,53970 2. IN DIVISION. Let it be required to find the quotient, which arifes by dividing one number by another ; suppose 1425 by 57, From From the logarithm of the dividend, viz. 1425=3,15381 Take the logarithm of the divifor, viz. 57 =1,75587 } rithm of the quotient, viz. 25=1,39794 3. In the Rule of Three. Three numbers given to find a fourth, in direa proportion. Rule. From the Tables take the logarithms of each of the propofed numbers, then, add the logarithms of the second and third together, and from the sum take the logarithm of the first, and the remainder will be the logarithm of the fourth number. Let the three proposed numbers be 18, 24 and 33, and the operation will ftand thus ; 1,38021=the logarithm of 24, the ad term. 2,89872=the logarithm of their product. -1,25527=the logarithm of the first term. 18. 1,64345=the logarithm of the 4th term required, which, by the Table, answers to the natural number 443 the 4th proportional to the three proposed numbers. 4. In INFOLUTION er RAISING POWERS. To find any power of any proposed number, or to involve any mumber to any propojed power, by logarithms. Rule.--Multiply the logarithm of the given root, by the power, viz. by 2 for the {quare, by 3 for the cube, &c. and the product is the Ingarithm of the power fought. Required to find the cube of i2 ? 1,07918=the logarithm of 12. X3=the third power, or cube. To Extract any Root of any proposed number. Rule.--Divide the logarithm of the proposed number by the index of the required root, viz. by 2 for the square, by 3 for the cube, &c. and the quotient will be the log: arithm of the root required. Requireet Required to find the cube root of 1728 ? 3,23754 = the logarithm of 1728, and 3,23754 3 = 1,07918 is the logarithm of the cube root of 1728, viz. 12. 6. In COMPOUND INTEREST. To find the amount of any sum for any time, and at any rate, at compound intereft. Rule.-Multiply the logarithm of the ratio (i. e. the amount of il. for 1 year) by the number of years, and to the product add the logarithm of the principal ; the sum will be the logarithm of the amount. What will 45l. amount to, forborne 12 years, at 6 per cent. per annum, compound interest ? Log. of 1,06, the ratio, is 902533 Multiply by the time 12 130396 Add log. of 45, the principal 1,65321 The sum is 1,95717 which is the logarithm of 90,7=£90 145. Anf. 7. In DISCOUNT, at COMPOUND INTEREST. To find the present worth of any sum of money, due any time bence, at any rate, at compound intereff: Rule. From the logarithm of the fum to be difcounted, subtract the logarithm of the rate multiplied by the time ; and the remainder the logarithm of the preient worth. What present money will pay a debt of gol. 145. due 1 2 years hence, discounting at the rate of 6 per cent, per: annum ? From the logarithm of £.90 14=1,95717 Sub. product of the Log. of the ratio X by the time = 930396 the} = The remainder 1,65421 is the logarithm of £45 Anf. N. Ham. |