6 Let a=the first term, then da=6, and a= also, since s= d (n.2a+n-1.d=) na+ .d by theor. 11. we have by sub 6x5 2 N.N- - 1 stitution, 48=6a+- .d, that is, 6a+ 15 d=48; whence 2a+ 6 16 5 d=16, or (putting — for a) 5 d2+12=16 d, or d2——d= 12 5 6 d 5 ; whence by completing the square, &c. d=2, therefore a= (=) 3; consequently the numbers are 3, 5, 7, 9, 11, and 13. 6. The continual product of four numbers in arithmetical progression is 380, and the sum of their squares 214; what are the numbers ? Let p 880, s=214, 2x=the common difference, y—3 x≈ the less extreme; then will y-3 x, y−x, y +-x, and y+3x=the terms of the progression: wherefore by the problem, y-3 x.y-x. y+x.y+3x=p, and y-3x2+y-x)2+y+x2+y+3x2=s; these equations reduced, become y+-10 y2x2+9x=p, and 4y2+20x2 $2 =s; from the latter of these y2=-5xo, therefore y1= 4 5 sx2 2 16 +25x+; if these values be substituted for their equals in the 16 x 84' 5 s R2 Р Р 52 4 84 -) x1—ax2 = · ; then by completing the square, &c. x= (by restoring the values of a and R) 1; whence y= (√5x2=) 64: therefore y-3x=2, y-x=5, y+x=8, and 4 y+3x=11, the numbers required. 20. To find the number of permutations, which can be made with any number of given quantities. Def. The permutations of quantities are the different orders in which they can be arranged. Let a and b be two quantities; these will evidently admit of two permutations, viz. ab and ba, which number of permutations may be thus expressed, 1×2. Let a, b, and c, be three quantities; these admit of six permutations, abc, bac, cab, acb, bca, and cba, viz. 1×2×3. Let a, b, c, and d, be four quantities; these admit of 24 permutations; thus, abcd bacd cabd dabc abdc badc cadb dacb That is, 4 things admit of 1x2×3×4 permutations. And therefore n things admit of 1x2x3, &c. to n, permutations. EXAMPLES.-1. How many ways can the musical notes ut, re, mi, fa, sol, la, be sung? Ans. 1×2×3×4x5×6=720 ways. 2. How many changes can be rung on 12 bells? Answer, 479001600. 3. How many permutations can be made with the 24 letters of the alphabet? 21. To find the number of combinations that can be made out of any given number of quantities. Def. The combinations of quantities, or things, is the taking a less collection out of a greater as often as it can be done, without regarding the order in which the quantities so taken are arranged. Thus, if a, b, and c, be three quantities, then ab, ac, and bc, are the combinations of these quantities, taken two and two: and here it is necessary to remark, that although ab and ba form two different permutations, yet they form but one combination; in the same manner ac and ca make but one combination, as also bc and cb. Let there be n things given, namely a, b, c, d, &c. (to n terms,) then if a be placed before each of the rest, n-1 permutations will be formed; if b be placed before each of the rest, n—1 permutations will in like manner be formed; and if c, d, e, &c. be placed respectively before each of the rest, n-1 permutations in each case will arise; consequently, if each of the n things be placed before all the rest, there will be formed in the whole n.n—1 permutations; that is, there can n.n—1 permutations be formed of n things taken two at a time. -1.n-2; Hence, if instead of n we suppose n— —1 things, b, c, d, e, &c. the number of permutations which these afford of the quantities taken two and two, will (by what has been shewn) be n— now if a be prefixed to each of these permutations, there will be n— 1.n—2 permutations in which a stands first; in the same manner it appears, that there will be n-1.n-2 permutations in each case when b, c, d, e, &c. respectively stand first; and therefore when each of the n things have stood first, there will be formed in the whole n.n—1.n-2 permutations of n things taken three and three. By similar reasoning it appears that n things taken 4 at a time afford n.n-1.n-2.n—3 5 at a time. r at a time.. ... n.n-1.n-2.n-3.n-4 n.n-1.n-2.n-3.n-4...n-r+1. permu Stations. This being premised, we may readily obtain the number of combinations, each consisting of 2, 3, 4, 5, &c. to r things, which can be made out of any given number n; for it appears by the preceding problem, that 2 things admit of 2 permutations, but by the definition they admit of but 1 combination; and therefore any number of things taken 2 at a time, admit of half as many combinations as there are permutations; but the number of permutations in n things, taken two and two, has been shewn to be n.n-1; therefore the number of combinations in n things, taken two and two, will be n.n-1 n.n-l or which is the same 1.2 and If three things be taken at a time, then 6 permutations will arise from every 3 things so taken, and but 1 combination ; therefore any number of things taken 3 at a time, admit of one sixth as many combinations, as there are permutations; but the number of permutations in n things taken 3 at a time, has been shewn to be n.n—1.n-2; and therefore the number of combina tions in n things, taken 3 at a time, will be N.N- -1.n-2 1.2.3 n.n-1.n or 6 By similar reasoning it may be shewn, that the number of combinations in n things, taken EXAMPLES.-1. How many combinations can be made of 2 2. How many combinations of 5 letters can be made out of the 24 letters of the alphabet? Here n=24, then n.n-1.n-2.n-3.n-4 1.2.3 4.5 =10626. Ans. 3. In a ship of war there are 40 officers, and the captain intends to invite 6 of them to dine with him every day; how many parties is it possible to make, so that the same 6 persons may not meet at his table twice? 22. To investigate the rules of simple interest. Def. 1. The sum lent is called the principal. 2. The money paid by the borrower to the lender for the use of the principal, is called interest. 3. The interest (or quantity of money to be paid) is previously agreed upon; that is, at a certain sum for the use of 1001. for a year: this is called the rate per cent. per annum ›. y Per cent. means by the hundred, and per annum, by the year; the term 5 per cent. per annum, means 5 pounds paid for the use of 100%. lent during the space of a year, &c. Various rates of interest have been given in this country for the use of 4. The principal and interest being added together, the sum is called the amount. Let p the principal lent, r=the interest of 1 pound for a year, t=the time during which the principal has been lent, i= the interest of p pounds for t years, a=the amount; then will 1 (pound): r (interest) :: p (pounds): pr=the interest of p pounds for a year and 1 (year): pr (interest) :: t (years): ptr=i (THEOR. 1.)=the interest of p pounds for t years, or t parts of a i i year: hence p· t= and r tr pr principal be added, we shall have ptr+p=a (THEOR. 2.) hence i If to this interest the pt and r= (THEOR. 5.) The following is a synopsis of the whole money, at different periods, from 5 to 50 per cent. but the law at present is, that not more than 5 per cent. per annum can be taken here, although the legal rate of interest is much higher in some of our colonies. The interest of money is computed as follows; In the courts of law .... in years, quarters, and days. .... On South Sea and India bonds ...... calendar months and days. On Exchequer bills.... quarters of a year and days. Brokerage, or commission, is an allowance made to brokers and agents in foreign, or other distant places, for buying and selling goods, and performing other money transactions, on my account; it is reckoned at so much per cent. on the money which passes through their hands, and is calculated by the rules of simple interest, the time being always considered as 1. The same rules serve for finding the value of any quantity of stock to be bought or sold, and likewise for finding the price of insurance on houses, ships, goods, &c. |