3. Divide the last product by the former, and the quctient will be the number sought. EXAMPLES. 1. How many combinations can be made of 6 letters out of ten ? 1x2x3x4x5x6(=the number to be taken at a time)=720 10x9x8x7x6x5(=same number from 10)=151200 720) 151200(210 the answer. 1440 2. How many combinations can be made of 2 letters out of 24 letters of the alphabet ? Ans. 276. 3. A general, who had often been succesful in war, was asked by his King, what reward he should confer on him for his services; the general only desired a farthing for every file of 10 men in a file, which he could make with a body of 100 men : what is the amount in pounds sterling? Ans. 180315723501. 9s. 2d. And the same thing will hold, let the number of things, to be taken at a time, be what it may; therefore the number of com binations of m things, taken n at a time, will X X 2. 3 PROBLEM 6. To find the number of combinations of any given number of thing's, by taking any given number at a time; in which there are several things of one sort, several of another, &C. RULE. 1. Find by trial the number of different forms, which the things, to be taken at a time, will admit of, and the number of combinations in each. 2. Add together all the combinations, thus found, and the sum will be the number required. EXAMPLES. 1. Let the things proposed be aaabbc; it is required to find the number of combinations, that can be made of every three of these quantities. 2. Let aaabbbcc be proposed; it is required to find the number of combinations of these quantities, taken 4 at a time? Ans. 10. 3. How many combinations are there in aaaabbccde, 8 being taken at a time? Ans. 13. 4. How many combinations are there in aaaaabbbbbccccdd ddeceefiffs, 10 being taken at a time? Ans. 2819. PROBLEM 7. To find the compositions of any number, in an equal number of sets, the things themselves being all different. RULE.* Multiply the number of things in every set continually together, and the product will be the answer required. * DEMONSTRATION. Suppose there are only two sets; then it is plain, that every quantity of one set, being combined with every quantity of the other, will make all the compositions of two things, in these two sets; and the number of these compositions is evidently the product of the number of quantities in one set by that in the other. Again, suppose there are three sets; then the composition of two, in any two of the sets, being combined with every quantity of the third, will make all the compositions of 3 in the 3 sets. That is, the compositions of 2 in any two of the sets, being multiplied by the number of quantities in the remaining set, will produce the compositions of 3 in the 3 sets; which is evidently the continual product of all the 3 numbers in the 3. sets. And the same manner of reasoning will hold, let the number of sets be what it will. Q. E. D. The doctrine of permutations, combinations, &c. is of very extensive use in different parts of the mathematics; particularly in the calculations of annuities and chances. The subject might have been pursued to a much greater length; but what has been done already.will be found sufficient for most of the purposes, to which things of this nature are applicable. EXAMPLES. 1. 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. 9 9 81 9 729 9 6561 Or, 9x9x9x9=6561 the answer. 2. Suppose there are 4 companies, in one of which there are 6 men, in another 8, and in each of the other two 9; what are the choices, by a composition of 4 men, one out of each company? Ans. 3888. 3. How many changes are there in throwing 5 dice? Ans. 7776. MISCELLANEOUS QUESTIONS. 1. WHAT difference is there between twice five and twenty, and twice twenty-five? Ans. 20. 2. A was born when B was 21 years of age; how old will A be when B is 47 ; and what will be the age of B when A is 60 ? Ans. A 26, B 81. 3. What number, taken from the square of 48, will leave 16 times 54 ? Ans. 1440. 4. What number, added to the thirty-first part of 3813, will make the sum 200 ? Ans. 77. 5. The remainder of a division is 325, the quotient 467, and the divisor is 43 more than the sum of both : what is the dividend ? Ans. 390270. 6. Two persons depart from the same place at the same time; the one travels 30, the other 35 miles a day : how far are they distant at the end of 7 days, if they travel both the same road ; and how far, if they travel in contrary directions ? Ans. 35, and 455 miles. 7. A tradesman increased his estate annually by 100l. more than 1 part of it, and at the end of 4 years found, that his estate amounted to 103421. 3s. 9d. What had he at first? Ans. 40001. 8. Divide 1200 acres of land among A, B, and C, so that B may have 100 more than A, and C 64 more than B. Ans. A 312, B 412, and C 476. 9. Divide 1000 crowns; give A 120 more, and B 95 less, than C. Ans. A 445, B 230, C 325. 10. What sum of money will amount to 1321. 16s. 3d. in 15 months, at 5 per cent. per annum, simple interest? Ans. 1251. 11. A father divided his fortune among his sons, giving A 4 as often as B 3, and C 5 as often as B 6; what was the whole legacy, supposing A's share 50001. ? Ans. 118751. 12. If 1000 men, besieged in a town with provisions for 5 weeks, each man being allowed 16oz. a day, were rein |