EXAMPLES. I. A lady purchased damask for a gown, at 8s. per ard, and lining for it at 3s. per yard; the gown and Ening contained 15 yards, and the price of the whole vas 31. 10s.-) -How many yards were there of each ? Suppofe 6 yards damaík, value 48s. Then she must have 9 yards of lining, value 275. Sum of their values=75s. So that the firft error is 5 too much, or + 5 Again, fuppofe fhe had 4 yards of damafk, value 32s. Then the must have 1 yards of lining, value 33s. Sum of their values=65s. So that the fecond error is 5 too little, or 65+ 55. X 4 5 Anf. 5 yards damask, and 15—5=10 yds. Or, 6+4÷2=5 as before. (lining. 2. A and B have the fame income: A faves of his, but B, by spending 30l. per annum more than A, at the end of 8 years finds himself 40l. in debt; what is their income, and what does each spend per annum ? Suppofe { 80 120+ Ans. Their income is 2001. 160 Then 80-10=70, A's expenfe per annum and Then 70+30=100, B's expenfe per annum. 100X8-80X8=160, which should have been 43 therefore 160-401 20 more than it should be, for th firit error In like manner proceed for the second error. 3. A and B laid out equal fums of money in trade: A gained a fum equal to of his flock, and B loft 2251; then 's money was double that of B; what did each lay out? √300 225+ Suppofe X 1900 225 Anf. 6001. 4. A labourer was hired for 60 days upon this condition that for every day he wrought, he fhould receive 3s. 4d and for every day he was idle, fhould forfeit is. 8d a the expiration of the time he received 31. 155. ; how many days did he work, and how many was he idle? Suppofe he worked 20 900 240 ~ 300+ Anf. He was employed 35 days and was idle 25 5. A gentleman has two horfes of confiderable valve, and a carriage worth tool. Now, if the firft horse be harneffed in it, he and the carriage together will be triple the value of the fecond; but if the fecond be put in, they will be 7 times the value of the firft? what is the value of each horfe? X Suppofe 44 160 Anf. One zol, and the other 4cl. 6. There is a fifh, whofe head is 10 feet long; his tail is as long as his head and half the length of his body, and his body as long as the head and tail? what is the whole length of the fifh? Head Head=10 First, fuppofe the body 20 to-Tail=30 2d, fuppofe it 30 X Body=40 which being increased by 7. What number is that, its, its 4 and 5 more, will be doubled? Suppofe { 8 3+ X 16 1+ Anf. 20. 8. A farmer having driven his cattle to market, received for them all, 8ol. being paid at the rate of 61. per ox, 41. per cow, and 11. 10s. per calf; there were as many oxen as cows, and 4 times as many calves as cows: How many were there of each fort? Suppofe cows 6 16+ Suppofe I 2 X Anf. 5 oxen, 5 cows, and 20 calves. 9. A, B and C built a ship which cost them £1000of which A paid a certain fum-B paid rool. more than A, and C 1ool. more than both; having finished her, they fixed her for fea, with a cargo worth twice the value of the ship: The outfits and charges of the voyage amounted to of the fhip; upon the return of which, they found their clear gain to be of of the vessel, cargo and expenfes : Pleafe to inform me what the hip coft them severally; what fhare each had in her, and what, upon the final adjustment of their accompts, they had severally gained? Suppofe it cost A £100 300- Suppofe 7 40 200 100+ A owned of the flip, which cost him 1751. and his fhare of the gain was 2181. 15s.-B owned. which coft 2751., and his gain was 3431. 15s-Cown ed, which coft 550l. and his gain was 6871. 10s. PERMUTATIONS F F PERMUTATIONS AND COMBNA TIONS. The Permutation of Quantities is the fhewing how many different ways any given number of things may be changed. This is alfo called variation, alternation, or changes and the only thing to be regarded here is the order they ftand in; for no two parcels are to have all their quentities placed in the fame fituation. The Combination of Quantitics is the fhewing how often a lefs number of things can be taken out of a great er, and combined together, without confidering their places, or the order they stand in. This is fometimes called election or choice; and here every parcel must be different from all the reft, and no two are to have precifely the fame quantitics or things. The Compofition of Quantities is the taking of a given number of quantities out of as many equal rows of different quantities, one out of every row, and combining them together. Here no regard is had to their places; and it differs from combination only, as that admits of but one row of things. PROBLEM I. To find the number of permutations, or changes, that can be made of any given wumber of things, all different from each other. RULE.* Multiply all the terms of the natural feries of numbers, from 1 up to the given number, continually together and the last product will be the anfwer required. EXAMPLES. Any two things and are capable of two variations only; as ab, ba; whole number is cxprcffed by 1×2. If there be three things, a.b and c, then any two of them, leaving out the third, will have 12 variations; and confequently when the third is taken in, there will be IX-Xj va riations; and to on, as far as you please. 1. EXAM PI, E S. Christ Church, in Bofton, has 8 bells; how many changes may be rung on them? 2. 1 X 2 X3 X4 X5×6×7×8=40320 the Anf. Nine gentlemen met at an inn, and were fo pleafed with their hoft, and with each other, that in a frolick they agreed to tarry fo long as they, together with their hoft, could fit every day in a different pofition at dinner pray how long, had they kept their agreement, would their frolick have lafted? Anf 99413 years. How many changes or variations will the alphaAnf. 620448401733 239 +39 360000. 3. bet admit of? 335 ; Any number of different things being given-to find how many changes can be made out of them, by taking any given number of quantities at a time. RULE. Take a feries of numbers, beginning at the number of things given, and decreafing by one, to the number of quantities to be taken at a time; the product of all the terms will be the anfwer required. EX A M P L E S. 1. How many changes may be rung with 4 bells out of 8 ? 8X7X6X5(=4 terms) 1680 the Anf. 2. How many words can be made with 6 letters of the alphabet, admitting a number of confonants may make a word? Anf. 96909120. Any number of things being given-whereof there are fev- 1. Take the feries 1X2 X3X4, &c. up to the * Any 2 quantities, a, b, both different, admit of two changes; but if the quantities are the fame, or ab become aa, there will be one alteration, which may be expreffed by: IX2 1X2=1. Any |