but these results (a) and (B) are equal to each other, for as is evident by clearing the equation of fractions; for then both sides are identical, but written in reverse order. Thus if n = = 8, and r = 3, 8.7.6 4.5.6.7.8 or the number of combinations of 8 quan1.2.3 5.4.3.2.1' then = tities, taken 3 together, is equal to the number of combinations taken 5 together, as is evident by bringing these fractions to a common deno minator. These combinations of n quantities, when taken r together, and n − r together, are said to be supplementary to each other. 145. Also if C, denote the number of combinations of n quantities taken r together, then the sum of all the combinations that can be made of n quantities, taken one, two, three, ... n together is n (n − 1) -- n (n = n + + 1.2 1) (n 1.2.3 2) 1. Find the number of permutations of the letters in the word change. Ans. 720. 2. Find the number of permutations of the letters in the word appliAns. 4989600. cation. 3. In how many ways may 7 persons seat themselves at table? Ans. 5040. 4. How many changes may be rung with 5 bells out of 8, and how many with the whole peal? Ans. 6720 and 40320. 5. The number of permutations of n quantities three together, is to the number of permutations five together, as 1 to 42; find n. Ans. n = 10. 6. The number of combinations of n quantities four together, is to the number two together, as 15 to 2; find n. Ans. n = 12. 7. On how many nights may a different guard be posted of 4 men out of a company of 36? and on how many of these will any particular soldier be on guard? Ans. 58905 and 6545. 8. If a company consisting of 30 men are drawn up in column, with how many different fronts can that be done when 5 men are always in front? Ans. 142506. 9. A captain, who had been successful in war, was asked what reward he expected for his meritorious services; he replied that he would be satisfied with a farthing for every different file of 6 men he could form with his company, which consisted of 100 men; what was the amount of his request? Ans. 12417211. 5s. 4 INTEREST AND ANNUITIES. 146. To find the amount of a given sum of money in any number of years, at simple interest. Let P denote the principal or sum at interest in pounds, r the rate of interest of 17. for one year, t the number of years the principal is at interest, and A the amount of principal and interest at the end of t years; then since r the interest of 11. for one year, tr the interest of 11. for t years, Ptr the interest of Pl. for t years, = .'. A=P+Ptr = P(1 + tr) From this equation, any three of the quantities P, r, t, A being given, the fourth may be found. 147. To find the amount of a given sum of money in any number of years, at compound interest. In addition to the notation in the former proposition, we shall make use of R to denote the amount of 17. for one year; then R = 1 + r. Now P in one year will amount to P+Pr = P (1 + r) = PR, and this being the principal for the second year, the corresponding amount will be PR + PR r = PR (1 + r) = P R2. In a similar manner PR is the amount in three years, and consequently in t years the amount will be (2), or, log A= log P+t log R. . . (2′). Cor. If the interest is paid half-yearly, then 2 t will be the number of payments, and the rate of interest; hence we have in this case 2 Hence we can find the time in which any sum at compound interest will amount to twice, thrice, or m times itself. Thus if A 2 P; then by (2) we have 2 if A or, if A = R', and t log R = log 2; 3P; then 3 R', and t log R = log 3; = m P; then m = R', and t log R = log m. = = EXAMPLES. 1. If 5007. be allowed to accumulate at compound interest at the rate of 57. per cent. per annum, what will be the amount at the end of 21 years? Ans. 13921. 19s. 74d. 2. In what time will any sum of money double itself at the rate of 4 per cent. per annum, compound interest being allowed? Ans. 15 7473 years. 3. If the population of a city contain one million of inhabitants, and increase at the average annual rate of 3 per cent., what will the population amount to at the end of 10 years? Ans. 1343915 inhabitants. ANNUITIES CERTAIN. 148. An annuity is a sum of money which is payable at equal intervals of time. When the possession of an annuity is not to be entered upon until the expiration of a certain period, it is called a reversionary or deferred annuity; and when the time of possession is not deferred, the annuity is said to be in possession. An annuity certain is one which is limited to a certain number of years; a life annuity is one which terminates with the life of any person, and a perpetuity, or perpetual annuity, is one which is entirely unlimited in its duration. 149. To find the amount of an annuity in any number of years, at compound interest. Let a denote the annuity, A the amount, R same as in the former investigations. = 1+r; r and n the The first payment becomes due at the end of one year, and if unpaid during the remaining n 1 years, it will amount in that time, at compound interest, to a R-1 pounds. The second payment becomes due at the end of two years, and, unpaid for n 2 years, will amount to a R"-2. In a similar manner, the third payment will amount in n-3 years to a R-3, and so on until the last payment which, unburdened with interest, is simply a. Hence the entire amount is the sum of a geometrical series, and therefore A = a + a R + a R2 + a R3 + be received in half-yearly instalments; then we have r)2” — 1 (1 + r)2 n If quarterly, A = a. 1 (1 + 4 r)'” − 1 Cor. If a pounds are placed out annually for n successive years, and the whole be allowed to accumulate at compound interest, then will 150. To find the present value of an annuity to be paid n years, at = compound interest. Let P denote the present value of the annuity a; then the amount of P pounds in n years PR" (147), and the amount of the annuity a in R" - 1 R-1 to each other; hence we get Cor. In the case of a perpetuity, n is infinite, and therefore we get 151. To find the present value of an annuity in reversion, commencing at the end of p years, and to continue q years. It is evident that if an annuity be deferred p years, and then continue q years, its present value will be less than that of an annuity to be received p + q years, by the present value of the annuity for p years; hence we have (150) If the annuity is payable for erer after the expiration of p years, then the value of the reversion of the perpetuity is (since q is infinite) 1. What is the present value of 3501., due at the end of 10 years, allowing 5 per cent. compound interest? Ans. 214l. 17s. 5d. 2. What is the value of a freehold estate, yielding an annual income of 2501., allowing 34 per cent. compound interest? Ans. 1421. 17s. 1d. 3. What will an annuity of 751. amount to in 6 years, at 4 per cent.? Ans. 4971. 9s. 4d. 4. A person has in perpetuity a property worth 5257. which he sells for 11666/. 13s. 4d.; what per cent. does the purchaser get for his money? Ans. 4 per cent. 5. What is the present value of an annuity of 1127. 10s., to commence at the end of 10 years, and to continue 20 years, at 4 per cent.? Ans. 10321. 17s. 6d. 6. If an annuity of 50l. be purchased for 613l. 18s. 34d., at 5 per cent. compound interest, what period must expire before the annuity is entered upon? Ans. 10 years. PROBABILITIES. 152. The probability of an event is the ratio of the number of chances for its happening to the number of chances both for its happening and failing. Thus if a expresses the number of favourable events and b the number of unfavourable events, the probability of its happening is the number of favourable events the whole number of events α a + b whilst the probability of its failure is expressed by b ; the number of unfavourable events the whole number of events a + b From this mode of representation it follows that certainty will be expressed by 1, and all probabilities will be expressed by fractions less than unity. The ratio of the probability of success to that of failure, or the ratio of the odds for or against, will be that of a to b, or of b to a. Hence, if 14 white and 10 black balls be thrown into an urn, the probability of drawing a white ball out of it, at one trial, is 10 14 and the 24' probability of failing or of drawing a black ball is 24 153. If a, a, be the number of ways in which two independent events may respectively happen, and b, b, the number of ways in which they may fail, then the probability that they will both happen is the product of the probabilities of the separate events, or a+b a+b1 a a (a+b) (a + b1)' For every case in a + b may be combined with every case in a, + b1, and thus form (a + b) (a, + b,) combinations of cases altogether; and each of the a cases in which the first event can happen may be combined with each of the a, cases in which the second event can happen, and thus form a a, combinations of cases favourable to the compound event; hence the probability that both will happen is (a + b) (a + b1)' (a + b) (a, + b1) (a + b) (a + b1) that the second will happen and the first fail is = ; The probability that the one will happen and the other fail, without specifying which event, is evidently the sum of the probabilities that the first will happen and the second fail, and that the second will happen and the first fail; hence it is 154. If a, a, a, be the number of ways in which three independent events may respectively happen, and b, b, b, the number of ways in which they may fail, then the probability that they will all happen is the continued product of the probabilities of the separate events, or a a + b a+b1 a 2 + b 2 = a a az (a + b) (a + b1) (as + b2)' The 'several combinations of all the cases in the first two chances, which by the last article are (a + b) (a, + b) in number, may be severally combined with the a, + b2 different cases of the third chance, and thus form (a + b) (a, + b,) (a + b2) combinations altogether. The favourable cases in the first two chances, which are a a, in number, may be combined severally with the a, favourable cases of the third chance, and thus form a a, a, favourable cases, and therefore the probability that all the three events will happen is the continued product of the simple chances, viz., The proposition may be extended to any number, n, of events, and the proof is similar to the preceding. Hence the probability that any number of events will all fail is equal to the product of all the separate probabilities of failing. VOL. I. |