By what rule do you work questions in Combination? Examples. 1. How many combinations of 3 persons in 6? 2. How many combinations of 10 figures may be made out of 20? Ans. 184756. Note.-The operation of a question may be contracted, by dividing the numbers of the increasing series into those of the decreasing, until the former be entirely cancelled. The continued product of the undivided numbers of the latter, and the quotients, will be the number of combinations; thus: 20. 19. 18. 17. 16. 16. 14. 13. 12. 11. 2 i. 2. 3. 4. $. %. *. %. p. 10. 19×2×17×13×2×11=184756. PERMUTATION. PERMUTATION is used to find how many different ways a given number of things may be varied in succession; as, 123, 132, 213, 231, 312, 321, are six different permutations of three figures. Rule. Multiply all the numbers continually in succession, from one to the given number inclusive; the product will be the number of variations. What is the rule for finding the number of variations in any given number ? Examples. 1. In how many different positions can 7 men place themselves round a table? 1X2X3X4X5X6X7=5040. Ans: 2. In what time will a person make all the changes that the 12 first letters of the alphabet admit of; allowing 15 seconds for each change? ·Ans. 227 years 248da. 6h. DUODECIMALS. DUODECIMALS are parts of a foot, the denominations of which increase continually by 12. What are Duodecimals? What are the denominations of Duodecimals? ADDITION OF DUODECIMALS. Rule. Add as in Compound Addition, and carry 1 for every 12 to the next denomination. 3. Three boards measure as follows: 16Ft. 8in., 14Ft. 6in., 17Ft. 9in. 2": how many feet do they contain? Ans. 48Ft. 11in. 2". SUBTRACTION OF DUODECIMALS. Perform the operation as in compound subtraction. 3. If, from a room measuring 475Ft. 7in. 2′′, I partition off 81 Ft. 2in. 5" 10" 6"", how long will the room still be? Ans. 394Ft. 4in. 8" 1"" 6"". MULTIPLICATION OF DUODECIMALS. Case 1. When the feet of the multiplier are not more than 12. Rule. 1 1. Set the multiplier in such a manner, that the feet thereof may stand under the lowest denomination of the multiplicand; and, in multiplying, carry 1 for every 12 from one denomination to another; and place the result of the lowest denomination in the multiplicand, under its multiplier. Case 2. When the feet of the multiplier exceed 12. Rule. Multiply by the component parts, as in compound mulLiplication; and take parts for the inches, as in practice. Questions. By what rule do you work, in multiplication of duodecimals, when the feet in the multiplier do not exceed 12? How do you proceed, when the feet of the multiplier exceed 12? 3. What are the contents of a door, measuring in length| 6Ft. 9in. 3", and in width 3Ft. 5in. Ans. 23 Ft. lin. 7" 3"". Case 2. 1. Multiply 208 Ft. 8in. 4", by 24Ft. 3in. 9". in. Ft. in." 2. A partition is 81 Ft. 10in. 4" long, and 14Ft. 7in. 5" high: how many yards does it contain? Ans. 132 Yd. 8ft. Tin. 9'' '''' 8'". 3. How many square feet of roof will 1000 shingles cover, when the shingles are 2Ft. 5in. 7" 2"" in length, and 5in. 3" 6" 5""" in width? Ans. 1088 Ft. 2in. 8" 3"" 3""".+ PROMISCUOUS EXAMPLES. 1. A. is 25 years old, B. 15 years older than A., and C. is 12 years older than B.: the ages of B. and C. are required. Ans. B. 40 years, C. 52 years. 2. A., B., and C. have $220 50cts., and are desirous to share it in the proportion of A. 1, B., and C. the rest; but B. is willing his share shall be divided equally between A. and C.: it is required, what will A., B., and C. receive individually, according to the first proportion; and what will be the shares of A. and C. each, after B.'s relinquishing his share? Ans. A. will receive $44 10cts., B. $36 75cts., C. $139 65cts.; A. will receive, after B. relinquishes, $62 47cts. 5m.; C. $158 02cts. 5m. 3. A person sells a piece of cloth at $56 25cts., and thereby loses 7 per cent.: what was the first cost? Ans. $60 81cts. 5m.+ 4. If A. travel by mail at the rate of 8 miles an hour, and when he is 50 miles on his way, B. start from the same place that A. did, and travel on horseback the same road at 10 miles an hour, how long and how far will B. travel to come up with A.? Ans. 25 hours, and 250 miles. 5. Bought a quantity of cloth for $750, of which I found to be inferior, which I had to sell at $1 25cts. per yard, and by this I lost $100: what must I sell the rest at per yard, that I shall lose nothing by the whole? Ans. $3 15cts. 6. If 1000 bricks lie 6 inches from each other, in a straight line, and a person be employed to gather them up, one by one, and place them on a pile, which is one foot from the first brick, how far will he have walked when he shall have placed the last brick on the pile? Ans. 94M. fur. 186yd. 2ft. 7. Three bricklayers, A., B., and C., can raise the walls of a house in 20 days; B., C., and D. in 24; C., D., and A. in 30; and A., B., and D. can do it in 36 days: in |