Against 15 in the table, and under 5 per cent., is 7.745, and this mul`tiplied by 637. gives 487.935 487l. 18s. 84d.* If the interest agreed on had been 6 or 8 per cent., the answers would have been 6.847 X 63431l. 7s. 2d. Or, 5.394 X 63 339l. 16s. 5d. Ex. 2. What ought to be given to a landlord for adding seven years, to a lease, of which fourteen years are unexpired, allowing the tenant 6 per cent. interest for his money, and the improved † rent to be 60l. per annum? CASE II. To ascertain the value of the fine which ought to be paid for renewing a given number of years in any lease. RULE. The value for renewing an additional term, or for adding any number of years to the unexpired part of an old lease, is equal to the difference between the value of the lease for the whole term, and the value of the unexpired part. Ex. 1. What ought to be given for the addition of seven years to a lease, of which 13 are unexpired; allowing 6 per cent. for money? The whole term for which the new lease is to be granted is 20 years; therefore, Table p. 220, under 6 per cent., and against 20 is 11.469, and against 13 is 8.852; therefore this last subtracted from the former, will leave 2.617 for the number of years' purchase which ought to be given for the renewal. Ex. 2. What should be given for the completing a 60 years' lease, of which a tenant has an unexpired term of 15 years, allowing him 7 per cent. for his money? NOTES. *This is the sum which, put out to interest at 5 per cent. would, after the next six years, the remainder of the lease, produce a clear annual income of 631. for 15 years; and therefore is the true sum that ought to be given for the advance of these 15 yearly rents of 631. each, and which the landlord would not otherwise receive till the end of the seventh and 14 following years. It often happens, that when a tenant applies for the renewal of the years lapsed in a lease, the estate has incseased in value since it has been in his possession; and in such cases the landlord usually demands a fine in proportion to what he conceives the rent ought to be from its increased value. This is called the improved rent. Ex. 3. I have a house for a lease of 48 years, but I wish to extend the lease to 97 years: how much must I pay for it, supposing the house worth 50l. per annum, and the interest 8 per cent? It will be seen, by working Ex. 2, of Case 1, by this rule, that the answer will be precisely the same by both methods: for the whole term for which the new lease is granted is 21 years; the value of a lease for this term is, by Table, p. 220, 11.764, and the value of the 14 years' lease yet to come is 9.295; this, subtracted from the other, gives 2.469, as before, which, multiplied by 60, and the answer is 148l. 2s. 9žd. The value of leases or estates for single or joint lives, or for the longest of two or three lives, is found by the same rules that have been given, p. 204-8, for finding the value of annuities for the same terms. When estates are held on two or three lives, and one of the lives nominated in the lease becomes extinct, the tenant is often desirous of replacing such life, or of putting in a new one, in order that the estate may continue to be held on the same number of lives in being, and thereby his interest in the same may be prolonged. In such cases it is customary, if the estate has improved in value since the original grant of the lease, for the landlord to demand a fine proportionate to such improved value; and to the age of the person intended to be added to those already in possession. The tenant will, as it is his interest, add one of the best lives he can find, that is, a life which has the greatest expectation of living, according to the best tables of mortality; and such a life will be about eight or ten years: at any rate few persons will be disposed to put in a life above the age of twenty. The following table will comprehend the cases that most frequently occur at the rate of 5 and 6 per cent. TABLE. For Renewing, with One Life, the Lease of an Estate held on Three RULE. The years' purchase in the table, multiplied by the improved annual value of the estate, beyond the rent payable under the lease, gives the fine to be paid for patting in the new life. Ex. What must be given to put in a life of 10 years, when the ages of those in possession are 40 and 50, allowing 6 per cent. for money? Answer, 2.204, or not quite 2 years' purchase. If the life to be added be 15 years, the answer would be 2.067, or very little more than 2 years' purchase. And, If the life to be added be 20 years, the answer would be 1.908, or less than 2 years' purchase. 5 PERMUTATIONS AND COMBINATIONS. THE PERMUTATION of quantities is the changing er varying the order of things. The COMBINATION of quantities is the shewing how often a less-number of things can be taken out of a greater, and combined together, without considering their places, or the order in which they stand. CASE I. To find the number of changes that can be made of any given number of things, all different from each other. RULE. Multiply all the terms one into another, and the last product will be the number of changes required. Ex. 1. How many changes can be rung on 12 bells? 1 X2 X3 X4 X5 X6 X 7 X 8 X 9 X 10 X 11 X 12479,001,600. Ex. 2. How many days can eight persons be placed in a different position at a dinner table? CASE II. Any number of different things being given, to find how many changes may be made out of them, by *taking a given number of quantities at a time. RULE. Multiply the number of things given by itself less 1, and that product by the same number less 2, diminishing each succeeding multiplier by an unit, till there are as many products, except one, as there are things taken at a time the last product will be the answer. Ex. How many changes can be rung with 4 bells out of 12? 12 X 12-1 X 12-2 X 12—3— 12 X 11 X 10 X 9 11880. Ex. 2. How many changes can be rung with 5 bells out of 10? Ex. 3. What number of words, containing each 6 letters, can be formed out of the 24 letters in the alphabet supposing any 6 to form a word? & CASE III. To find the combinations of a less number of things out of a greater, all different. RULE. Take the series 1, 2, 3, 4, &c. up to the less numher of things, and multiply them continually together for a divisor: then take a series of as many terms, decreasing, PERMUTATIONS AND COMBINATIONS. 229 each by an unit, from the greater number of things, and multiply them continually together for a dividend. Divide the latter product by the former, and the quotient will be the answer. Ex. 1. How many combinations can be made of 10 things out of 100? 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 (the number to be taken at a time) 3,628,800. -100 X 99 X 98 X 97 X 96 X 95 X 94 X 93 X 92 X 91 and 62,815,650,955,529,472,000. 62815650955529472000 3628900 17310309456440.* Ex. 2. How many combinations can be made of 3 letters out of the 24 letters in the alphabet? Ex. 3. A club of 21 persons agreed to meet weekly, five at a time, so long as they could, without the same five persons meeting together, how long would the club exist? CASE IV. To find the compositions of any number, in sets of eqnal numbers, the things or persons themselves being different. RULE. Multiply the number of things in every set continually together, and the product is the answer. Ex. 1. There are three parties of cricketters, in each eleven men, in how many ways can 11 of thein be chosen, one out of each? Answer, 11 X 11 X 11 1331. Ex. 2. In how many ways can the four suits of cards be taken, four at a time? Ex. 3. There are four parties of whist-players; in one there are 6, in the second 5, in the third 4, and in the fourth 3 persons, how often can the set differ with these persons? NOTE. 10 X 11 X 49 X 12 X 19 X 31 X 23 X 13 X.94 X * Operations of this sort are shortened by the following mode : as above; and dividing the 12 by 6, we place 2 among the numerators, and get rid of all the denominators. |