4. What number of variations will the 9 digits admit of? Ans. 362880. 5. How many changes may be made on the 26 letters of the alphabet? Ans. 403,291,461,126,605,635,584,000,000. Note. From the answer to this last question, which amounts to a number, of which we cannot form any conceivable idea, we may discover the surprising power of numbers, and also the endless variety of ideas that may be distinctly communicated by these 26 simple characters. It will also be evident from the method of notation here used, that a row of figures of any given length whatever, may be numerated, though we may be entirely unable to comprehend the amount. SECTION 10. COMBINATION. COMBINATION of quantities, is the showing how often a less number of things can be taken out of a greater, and combined or joined together differently. RULE. Take a series of 1, 2, 3, &c. up to the number to be combined; take another series of as many places, decreasing by unity from the number out of which the combinations are to be made; multiply the first continually for a divisor, and the latter for a dividend, and the quotient will be the answer. EXAMPLE. 1. How many combinations may be made of 7 dollars out of 12? .1×2×3, &c. up to 7=5040 divisor. Again, 12 the whole number of terms less 7=5 2. How many combinations can be made of 6 letters out of 24 of the alphabet? Ans. 134596. 3. In how many different ways may an officer select 8 men out of 30, so as not to make the same selection twice Ans. 5852925 PART VIII. MENSURATION. SECTION 1. Duodecimals, or Cross Multiplication. DUODECIMALS are fractions of a foot, or of an inch, or parts of an inch, having 12 for their denominator. Inches and parts are sometimes called primes ('), seconds (''), thirds ('''), &c. Note. This rule is much used in measuring and computing the di mensions of the several parts of buildings; it is likewise used to find the tonnage of ships, and the contents of bales, cases, boxes, &c. ADDITION OF DUODECIMALS. RULE. Add as in compound addition, carrying 1 for each 12 to the next denomination. 3. From a board measuring 35 feet, 9 inches, 2 seconds, cut 24 feet, 10 inches, 5 seconds, and 4 thirds; what is left? Ans. 10ft. 10in. 8sec. 8''' 4. A joiner having lined several rooms very curiously with costly materials, finds the amount to be, in square measure, 803 fect, 3 inches, 4 seconds; but several deduc. tions being to be made for windows, arches, &c. those de ductions amounted to 70 feet, 3 inches, 7 seconds, 10 thiro 5 fourths; how many feet of workmanship must he be paid for? Ans. 732ft. İlin. 8'' 1''' 7'''' MULTIPLICATION OF DUODECIMALS. Case 1. When the feet of the multiplier do not exceed 12. RULE. Set the feet, or the highest denomination of the multiplier under the lowest denomination of the multiplicand, and multiply as in compound numbers, carrying 1 for every 12 from one denomination to another, and place the result of the lowest denomination in the multiplicand under its multiplier. TABLE. 1. Feet multiplied by feet, give feet. 2. Feet multiplied by inches, give inches. 3. Feet multiplied by seconds, give seconds, &c. 5. Inches multiplied by seconds, give thirds, &c. 6. Seconds multiplied by seconds, give fourths. 7. Seconds multiplied by thirds, give fifths, &c. PROOF. Reduce the given sum to a decimal, or work by the rules 5. What is the price of a marble slab, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches, at 1 dollar and 50 cents per foot? Ans. 15 dolls. 35,5 cts. 6. There is a house with three tiers of windows, 3 in a tier, the height of the first tier is 7 feet 10 inches, of the second 6 feet 8 inches, and of the third 5 feet 4 inches, and the breadth of each window is 3 ft. 11 inches.; what will the glazing come to at 14 cts. per ft.? Ans. 32 dolls. 62 cts. |