d, we can make six combinations of two letters out of the four, since the new quantity d, may be combined with each of the former ones; thus, a b, a c, b c, a d, bd, cd; and this number of combinations may be expressed 47.3 If we would make a combination of four, it is evident that only one such combination can be made out of the letters á b c d; but if a fifth letter, e, be added, we can make five such combinations; thus, a b c d, a b c e a bed, a ecd, b c d e; and this number of combina 5 X 4X3X2 tions may be expressed by 1x2x384 PROBLEM I. To find the number of combinations from any given number of things, all different from each other, taking a given number at a time. RULE. Take a series of numbers, the first term of which is equal to the number of things out of which the combinations are to be made, and decreasing by 1, till the number of terms is equal to the number of things to be taken at a time, and find the product of all the terms. Then take the natural series 1, 2, 3, fc. up to the number of things to be taken at a time, and find the product of all the terms of this series. Divide the former product by the latter, and the quotient will be the answer. 1. How many combinations of 3 letters can be made out of the 6 letters a b c d e f? 2. How many different yoke of oxen may be selected from twelve oxen ? 3. How many span of horses can be selected from eighteen horses ? 4. A drover agreed with a farmer for a dozen sheep, to be selected out of a flock of two dozen; hut while he was making the selection, the farmer told him, he might take the whole flock, if he would give him a cent for every different dozen that could be selected from it. To this the drover readily agreed. How many dollars did the whole flock come to, at that rate? 5. A general, who had often been successful in war was asked by his king, what reward he should confer upon 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. How much did the general's modest request amount to? PROBLEM II. To find the various combinations of a given number of things, which may be made out of an equal number of sets of different things, one from each set. RULE. Multiply the number of things in the several sets continually together, and the product will be the answer. A combination of this kind is called the composition of quantities. The rule may be illustrated thus. If there are only two sets, and we combine every quantity of one set with every quantity of the other set, we shall make all the compositions of two things in these two sets; and the number of compositions is evidently the product of the number of things in one set by the number of things in the other set. Again, if there are three sets, then the compositions of two in any two of the sets, being combined with every quantity of the third set, will make all the compositions of three in the three sets. That is, the compositions of two in any two of the sets, being multiplied by the number of things in the third set, will give all the compositions of three in the three sets; and this result is the joint product of all the numbers in the three sets. 6. Suppose there are four companies, in each of which there are 9 men; in how many ways can 4 men be chosen, one out of each company y? 7.- Suppose there are five parties, at one of which there are 6 young ladies, at another 8, at a third 5, at a fourth 7, at a fifth 10. How many choices are there, in selecting 5 young ladies, one from each party? 8. How many changes are there in throwing four dice, each die having six sides? 9. A certain farmer has 5 barns, in one of which he has 15 cows, in another 11, in another 5, in another 2, and in another 7. How many different selections may be made, in choosing 5 cows, one from each barn? 10. How many variations can be made in selecting å flock of a dozen sheep from 12 folds, one from every fold, in each of which there are 10 sheep? 11. In a certain school there are seven classes, the first containing 12 boys, the second 7, the third 9, the fourth 10, the fifth 11, the sixth 8, and the seventh 13. How many variations can be made in selecting 7 boys, one from each class? XXXVIII. EXCHANGE. Scholars, who are to prosecute a course of classical studies, and those, who are not expected to engage in any extensive mercantile business, máy omit the exercises in this article. Exchange is the act of paying or receiving the money of one country for its equivalent in the money of another country, by means of Bills of Exchange. This operation, therefore, comprehends both the reduction of moneys and the negotiation of bills; it determines the comparative value of the currencies of different nations, and shows how foreign debts are discharged, and remittances made from one country to another, without the risk, trouble, or expense of transporting specie or bullion. A Bill of Exchange is a written order for the payment of a certain sum of money, at an appomed time. It is a meri antile contract, in which four persons are mostly concerned, as follows. First- The Drawer, who receives the value, and is also called the maker and seller of the Bill. Second-The debtor in a distant place, upon whom the Bill is drawn, and who is called the Drawee. He also is called the Acceptor, after he accepts the Bill, which is an engagement to pay it when due. Third— The person who gives the value for the Bill, who is called the Buyer, Taker and Remitter. inent it to any Fourth--The person to whom the bill is ordered to be paid, who is called the Payee, and who may, by endorsepass other person. Many mercantile payments are made in Bills of Exchange, which pass from hand to hand, until due, like any other circulating medium; and the person who at any time has a Bill in his possession, is called the holder. To transfer a Bill payable to order, the payee should express his order of paying to another person, which 19 always done by an endorsement on the back of the Bill. An endorsement may be blank or special. A blank endorsement consists only of the endorser's name, and the Bill then becomes transferable by simple delivery. A special endorsement orders the money to be paid to a particular person, who is called the endorsee, who must also endorse the Bill, if he negotiates it. A blank endorsement may always be filled up with any person's name, so as to make it special. Any person may endorse a Bill, and every endorser, as well as the acceptor, is a security for the Bül, and may be sued for payment. In reckoning when a Bill, payable after date, becomes due, the day on which it is dated, is not included. When the time is expressed in months, calendar months are understood; and when a month is longer than the succeeding, it is a rule not to go, in the computation, into a third month. Thus, if a Bill be dated the 28th, 29th, 30th, or 31st, of January, and payable one month after date, the term equally expires on the last day of February. An endorsement inay take place at any time after the Bill is issued, even after the day of payment is elapsed. When the holder of a Bill dies, his executors may endorse it; but, hy so doing, they become answerable to their endorsee personally, and not as executors. A Bill payable to bearer is transferred by simple delivery, without any endorsement. Bills should be presented for acceptance, as well as for payment, during the usual hours of business. The common way of accepting a Bill is for the drawee to write his name at the bottom or across the body of it, with the word, accepted. When acceptance or payment has been refused, the Holder of the Bill should give regular and immediate notice to all the parties, to whom he intends to resort for payment; for if he do not, they will not be liable to pay. With respect to the manner, in which notices of nonacceptance or non-payment are to be given, a difference exists between Inland and Foreign Bills. In the case of Foreign Bills, a Protest is indispensably necessary: thus, a Public Notary appears with the Bill, and demands either acceptance or payment (as the case may be;) and on being refused, he draws up an instrument, called a Protest, expressing that acceptance or payment (as the case may be) has been demanded and refused, and that the holder of the Bill intends to recover any damages which he may sustain in consequence. This instrument is admitted, in foreign countries, as a legal proof of the fact. The Protest on a Foreign Bill should be sent as soon as possible, to the drawer or negotiator; and if it be for non-payment, the Bill must be sent with the Protest. A Protest is not absolutely necessary to entitle the holder to recover the amount of an Inland Bill from the drawer or endorser: it is sufficient if he give notice, by letter or otherwise, that acceptance or payment (as the case may be) has been refused, and that he does not mean to give credit to the drawee. If the person, who is to accept, has absconded, or cannot be found at the place mentioned in the Bill, Protest is to be made, and notice given, in the same manner as if acceptance had been refused. It is customary, as a precaution against accident or miscarriage, to draw three copies of a Foreign Bill, and to send them by different conveyances. They are denominated the First, Second, and third of Exchange; and when any one of them is paid, the rest become void. When acceptance is refused, and the Bill is returned by Protest, an action may be commenced immediately against the Drawer, though the regular time of payment be not arrived. His debt, in such case, is considered as contracted the moinent the Bill is drawn. |