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so continuing the multiplications as many times as there are things to be taken at a time; and the things to be taken at a time are 3; we therefore find the continued product of the first three numbers, beginning at 4; thus,
4 X 3 X 2= 24 Answer. NOTE. - The questions under this rule refer to permutations and not combinations. 5. How many changes can be rung with 4 bells out of 8?
OPERATION. 8 X 7 X 6 X 5= 1680 changes, Answer. 6. How many words can be made with 6 letters out of 26 of the alphabet, admitting a word might be made from consonants ?
CASE III. To find the number of combinations of any given number of things, all different one from another, taking any given number at a time, without reference to their arrangement.
RULE. - Take the series 1, 2, 3, 4, 8c., up to the number to be taken at a time, and find the product of all the terms.
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.
Divide the last product by the former, and the quotient will be the number required.
7. How many combinations can there be of any three letters of the alphabet out of any four letters, without reference to their arrangement or permutations ?
Illustration. — By examining four letters, a, b, c, d, we find they will admit of only four combinations. Thus, abc, abd, acd, bcd. And this number can be ascertained in the following manner:-1 X 2 X 3= 6; 4 x 3 x 2 = 24; 24 6= 4, the number of combinations.
8. How many combinations can be made of 7 letters out of 10, the letters all being different ?
Ans. 120. OPERATION. 1X2 X 3 X 4 X 5 X 6 X 7= 5040; 7 being the number taken at a time.
10 X 9 X 8 X7 X 6 X 5 X 4= 604800 = same number frɔm 10.
604800 - 5040= 120 Ans. Number of combinations.
CASE IV. To find the compositions of any number of things, in an equal number of sets, the things being all different.
Rule. — Multiply the number of things in every set continually together, and the product will be the answer required.
9. Suppose there are 4 companies, in each of which there are 9 men; it is required to find how many ways 4 men may be chosen, one out of each company. 9x9x9x9=6561 Ans.
10. There are 4 companies, in one of which there are 6 men, in another 8, and in each of the other two 9 men. What are the choices, by a composition of 4 men, one out of each company?
Ans. 3888. 11. How many changes are there in throwing 5 dice ?
INSURANCE on life is a contract, which stipulates for the pay. ment of a certain sum of money on the death of an individual, in consideration of the immediate payment of a specified sum, or more frequently of an annuity, or annual premium, to be continued during the existence of the life insured. Contracts of this kind are of immense importance to society. Every man whose income depends on his own life or exertions, and on whom others are dependent for support, must be sensible of the advantages of arrangements by means of which, at a small sacrifice of immediate comfort, he is enabled effectually to provide against the casualties of life. Though nothing can be more uncertain than the continuance of an individual life, yet nothing is more invariable than the duration of life in the mass ; consequently, the exact value of life insurances can be calculated without any uncertainty whatever, and a man by effecting an insurance secures to his family, against risk of accident, the advantages they would have from enjoying his exact proportion of the average duration of life. Such transactions provide against destitution, and tend directly to the accumulation of capital. No wise man will therefore neglect to provide against contingencies.
The following table from Milne's Treatise on the Valuation of Annuities and Assurances (Vol. II. p. 565) shows the “ expectation of life" at every age, according to the law of mortality at Carlisle.
Expectation of Life. Expectation. Age. Expectation. || Age. Expectation.|| Expectation. 38.72 26 37.14
3.37 15 45.00
7.72 99 2.77
7.33 100 2.28
7.01 101 1.79 24 38.59 50 21.11
6.69 102 1.30 25 37.86
20.39 77 6.40 103 0.83
By the above table it will be perceived, that the average age to which 100 persons will live from birth will be 387% years, and that 100 persons having attained the age of 35 years will live on an average 31 years longer ; and the average time which 100 persons will continue to live, after the age of 100 years, is 2-28 years. Taking the above table as the basis of their calculations, various life-insurance companies have been formed, whose tables or rates of insurance vary according as they charge a greater or less per cent. on their capital. We have selected two tables from those used by the numerous corporations which have been formed in the United States. The first is the one adopted by the Massachusetts Hospital Life Insurance Company, and the other by the New York Life Insurance Company.
.95 .97 .99 1.03 1.01 1.12 1.17 1.23 1.28 1.35 1.36 1.42 1.46 1.48 1.50
1.01 1,03 1.05
1.76 1.83 1.88 1.89 1.92 1.94 1.96 1.98 1.99 2.02 2.04 2.09 2.20 2.37 2.59 2.89 3.21 3.56 4.20 4.31 4.63 4.91
2.57 2.64 2.75 2.81 2.90 3.05 3.11 3.20 3.31 3.40. 3.51 3.63 3.3 3.87 4.01 4.17 4.49 4.60 4.75 4.90 5.24 5.49 5.78 6.05 6.27 6.50 6.75 7.00
The preceding table shows at what rate a hundred dollars may be insured on a person's life for one year, for seven years, ten years, and for life, at any age from 14 to 60 years. Thus, a man who is 42 years old, and who wishes to obtain a life insurance in Massachusetts for one year, for $ 100, must pay $1.31, and in the same proportion for a larger sum. And, if he would obtain a life insurance in New York, he must pay annu. ally $3.40 on every hundred dollars for which he obtains insurance.
EXAMPLES. 1. What premium will the Massachusetts Hospital Life Insurance Company require for the insurance of a life for 1 year for $1728, the person being 30 years of age ? Ans. $17.45.
$ 1728 x .0101 = $ 17.45 Ans. Note. -As the premium is $ 1.01 on $ 100, the sum insured must be multiplied by 100 = .0101.
2. What amount of premium must James Kimball pay at the above office to effect an insurance on his life for 7 years for $ 8000, his age being 53 years ?
Ans. $ 220. OPERATION. $ 8000 X.0275= $ 220 annually, Ans. 3. John Smith, 60 years of age, wishes to engage in a very profitable speculation, and, being destitute of the necessary funds, he effects an insurance on his life for 10 years, for $78,000, at the office of the above Company. Required the amount of the annual premium.
Ans. $ 3876.60. 4. What will be the premium per annum for insuring a person's life, who is 15 years old, for $ 2000 for 7 years, at the New York Life Insurance Company ? - Ans. $ 17.60.
5. A gentleman 45 years of age, being bound on a long and dangerous voyage, and wishing to secure a competence for his family, obtains an insurance for life at the above office in New York, for $ 12,000. By an act of Providence, he dies before the end of the third year. What is the net gain to his family?
Ans. $ 10,657.20. 6. John Swan, 20 years old, effects an insurance for life at New York, for $10,000, for which he pays an annual premium of 177 per cent. If the Insurance Company loan the premium at 6 per cent. compound interest, and Swan should die at the age of 60 years, who will gain by the insurance ?
Ans. Company will gain $17,392.86+. Note. — All premiums are paid annually and in advance.