CHAPTER XIV. PERMUTATION AND COMBINATION. PERMUTATION shows the number of changes that can be made in the order of a given number of things. PROBLEM I. To find the number of changes that can be made of any given number of things, all different from each other. How many changes may be made in the position of 4 persons at table ? If there were but two persons, a and b, they could sit in but two positions, ab and ba. If there were three, the third could sit at the head, in the middle, or at the foot, in each of the two changes, and there could then be 1x2x3=6 changes. If there were 4, the fourth could sit as the 1st, 2d, 3d, or 4th, in each of these 6 changes, and there would then be 1x2x3x4=24 changes. RULE. Multiply together the series of numbers 1, 2, 3, &c., up to the given number, and the product will be the number sought. 1. How many variations may be made in the order of the 9 digits ? 2. How many changes may be made in the position of the letters of the alphabet ? 3. How long a time will be required for 8 persons to seat themselves at table in every possible order, if they eat 3 meals a day? PROBLEM II. Any number of different things being given, to find how many changes can be made out of them by taking a given number of the things at a time. If we have five things, each one of the 5 may be placed before each of the others, and we thus have 5 X 4 permutations of 2 out of 5. 'If we take 3 at a time, the third thing may be placed as 1st, 2d, and 3d, in each of these permutations, and we have 5 X 4x3 permutations of 3 out of 5. For a similar reason we have 5x4x3x2 permutations of 4 out of 5, &c. RULE. Take a series of numbers, commencing with the number of things given, and decreasing by 1, until the number of terms is equal to the number of things to be taken at a time: the product of all the terms will be the answer required. 4. How many changes can be rung with 8 bells, taking 5 at a time ? 5. How many numbers of 4 different figures each, can be expressed by the 9 digits ? 6. In how many different ways may 10 letters of the alphabet be arranged ? PROBLEM III. To find the number of permutations in any given number of things, among which there are several of a kind. How many permutations can be made of the letters in the word terrier ? If the letters were all different, the permutations, according to Problem I. would be 1x2x3x4X 5 X 6X7=5040. But the permutations of the three r's would, if they were all different, be 1x2x3, which could be combined with each of the other changes; the number must therefore be divided by 1x2x3. For the same reason it must also be divided by 1x2, on account of the 2 e's. Then the true number sought is 1X2 X3 X4 X5 X 6X7 —420. 1X2 X3X1X2 RULE. Take the natural series, from 1 up to the number of things of the first kind, and the same series up to the number of things of each succeeding kind, and form the continued product of all the series. By the continued product divide the number of permutations of which the given things would be capable if they were all different, and the quotient will be the number sought. 7. How many changes can be made in the order of the letters in the word Philadelphia ? 8. How many different numbers can be made, that will employ all the figures in the number 119089907343 ? 9. How many permutations can be made with the letters in the word Cincinnati ? COMBINATION shows in how many ways a less number of things may be chosen from a greater. If we have ten articles, each may be combined with every one of the nine remaining ones, and therefore we may have 10x9 permutations of 2 out of 10. But each combination will evi dently be repeated; thus, we shall have ob and ba, ac and ca, &c. 10x9 Therefore, the number of combinations will be. 2 If now we add an eleventh article, each of the eleven may be joined to each of the combinations of the remaining ten, and we 11 X 10X9 shall have permutations. But each combination will 1X2 be three times repeated; thus we shall have abc, bac, and cab; abd, bad, and dab, &c. The number of combinations of 3 out of 11 will therefore be 11 X 10 X9 Hence we obtain the fol 1X2 X3 lowing RULE. Write for a numerator the descending series, commencing with the number from which the combinations are to be made, and for a denominator the ascending series, commencing with 1, giving to each series as many terms as are equivalent to the number in one combination. Cancel the like factors in the numerator and denominator, and divide. 10. How many combinations of 4 letters, can be made from the alphabet ? 11. How many combinations of 7 can be made from 18 apples? 12. How many ranks of 10 men, may be made in a company of 80 ? 13. How many locks of different wards, may be unlocked with a key of 6 wards ? [Find the number of combinations of 1, 2, 3, 4, 5, and 6 in 6, and the sum of all the combinations will be the number required.] CHAPTER XV. IN VOLUTION. INVOLUTION is the repeated multiplication of a number by itself. The product obtained by Involution is called a power. The root is the number involved, or the first power. If the root be multiplied by itself, or employed twice as a factor, the product is the second power. If the root is employed three times as a factor, it is raised to the 3d power ; if 5 times, to the 5th power, &c. Thus, 2 is the 1st power of 2, or 21. 2x2 or 4, is the 2d power of 2, or 22. 2x2x2 or 8, is the 3d power of 2, or 23. 2x2 x 2x2x2 or 32, is the 5th power of 2, or 25. The power is usually denoted by a small figure over the right of the root, called the exponent, or index. When there is no exponent, the number is regarded as the 1st power. The second power is often called the square, because the number of square feet in any square surface, is obtained by multiplying the number of feet in one side by itself. The third power is often called the cube, because the number of cubic feet in any cubical block, may be ob. tained by raising the number of feet in one side to the 3d power. The 4th power is sometimes called the bi-quadrate, or the square squared; the 5th power, the first sursolid ; the 6th power, the square cubed, or the cube squared; the 7th power, the second sursolid; the 8th power, the bi-quadrate squared ; the 9th power, the cube cubed ; the 10th power, the 1st sursolid squared, &c. If the exponents of any two powers of the same number be added, we shall obtain the exponent of their product. Thus 63 x 65=6x 6 x 6 x 6x6x6 x 6x6 = 68 ; 42 x 43=4X4X4X4 X4=45. In any two powers of the same number, if we subtract the smaller exponent from the larger, we shall obtain their 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 quotient. Thus 68:-66= 6 X 6 X 6 X 6 X 6 6x6x6-63. We may represent any power of a number, by multiply. ing its exponent. Thus, the 7th power of 5 is 57; the 3d power of 22 is 26, because 22 x 22 x 22=26. These properties form the basis of the system of Logarithms. 1. What is the 2d power of 6 ? the 3d power ? 5. What is the difference between 34 and 43 ? 6. What is the value of 117; 37; 23 x 22 ? 7. What power of 9 is equivalent to 95 x 93 ; 92 x 910; 94 x 96 ; 9x 97 x 98 ? 8. Multiply 1279 by 1277, and divide the product by 12715. 9. Divide 31 9 by 319 ; 178 by 175; 427 by 426. 10. What is the sixth power of 41 ? 11. What is the 9th power of 53 ? the 12th power of 185 ? the 24th power of 172 ? CHAPTER XVI. EVOLUTION. EVOLUTION is the process by which we discover the root of any given power. Thus, 3 is the 2d root of 9, the 3d root of 27, the 5th root of 243, because 9=32, 27=33, 243=35. So the 2d or square root of 49 is 7; the 3d or cube root of 125 is 5; the 4th root of 16 is 2 ; the 5th root of 1024 is 4, &c. We may denote a root by a radi. cal sign, or by a fructional exponent. The radical sign is ✓ , and when employed by itself denotes the square root. If we wish to denote the 3d, 5th, 7th, &c. root, the index of the root is written above the radical sign thus, 3 5+, &c. In fractional exponents, the numerator expresses the power of the number, and the denominator expresses the root. Thus, (27)} = 3/27; (16)*= »163 ; (32)# = 3324, 8&c. The product, or the quotient, of two second, third, or other roots, is the 2d, 3d, &c., root of the product or quotient. Thus, 327x 125=127x 125 or V3375. For 27=33=3x3x3, and 125=5* 5 5. Then 27 x 125= 3x3x3x5x5x5=3*5*3*5*3*5=153. Therefore 727X 125=15. In a similar manner it may be shown that V 3375:- 3125= { 27. |