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3. How many different permutations may be made of 10 things taken all together?
Ans. 3628800. 4. How many different numbers can be formed with the five Arabic characters, 4, 3, 2, 1, 0; each of the characters occurring once, and only once in each number?
Ans. 96. 5. How many different combinations may be formed of 8 things taken 4 at a time?
6. How many different combinations may be made of 16 things taken 5 at a time?
7. How many different parties of 6 men each can be formed from a company of 20 men ?
Ans. 38760. 8. In how many different ways can a class of 6 boys be placed in line, one boy being denied the privilege of the head ?
Ans. 600. 9. Find the greatest number of different products that can be formed with the prime numbers under 40, the products being all composed of the same number of factors.
Ans. 1716. 10. The number of permutations of n things taken 5 at a time, is equal to 120 times the number of combinations of the n things taken 3 at a time; find n.
Ans. n = 8. 11. At a certain house there were 8 regular boarders; and one of them agreed with the landlord to pay $35 for his board so long as he could select from the company different parties, equal in number, to sit each for one day on a certain side of the table. At what price per day did he secure his board ?
Ans. $.50. 12. A and B have each the same number of horses; and A can make
different teams by taking 3 horses together, as B can by taking 2 together. Required the number of horses that each has.
twice as many
13. There are 12 points in a plane, no three of which are in the same straight line with the exception of five, which are all in the same straight line. How many different straight lines can be formed by joining the points.
342. A Series consists of a number of terms following one another, but so related that each may be derived from one or more of the preceding, by a fixed law.
A series may be finite or infinite, converging or diverging.
343. A Finite Series is one which by its law of development must terminate, or have only a finite number of terms.
344. An Infinite Series is one which by the law of its development can never terminate, but may have an infinite number of terms.
345. A Converging Series is one whose successive terms con tinually diminish in numerical value.
346. A Diverging Series is one whose successive terms continually increase in numerical value.
347. An Arithmetical Progression is a series of numbers or quantities increasing or decreasing from term to term by a common difference.
We may consider the common difference as a quantity continually added, in the algebraic sense ; hence, it will be positive in an increasing series, and negative in a decreasing series. Thus,
1, 3, 5, 7, 9,.... is an increasing arithmetical progression, in which the common dif. ference is +2; and
20, 18, 16, 14, 12,.... is a decreasing arithmetical progression, in which the common dif. ference is 2.
348. To investigate the properties of an arithmetical progres. sion, we may suppose the series to terminate ; there will then be five parts or elements ;—the first term, the last term, the number of terms, the common difference, and the sum of the terms. The first term and last term are called the extremes, and all the terms between the extremes are called arithmetical means.
349. In an Arithmetical Progression, the last term is equal to the first term plus the common difference multiplied by the number of terms less 1.
Let a denote the first term, l the last term, d the common difference, and n the number of terms; then the series will be represented thus :
a, (a+d), (a+2d), (a +3d),..... And we perceive that in every term the coefficient of d is equal to the number of preceding terms; hence,
l=a+(n-1)d, in which d is positive or negative, according as the series is an increasing or a decreasing one.
350. In an arithmetical progression the sum of any two terms equidistant from the extremes is equal to the sum of the extremes.
Let t denote a term of the series which has r terms before it, and t' a term which has r terms after it; then the terms, t and ť, will be equidistant from the extremes. Suppose the series to be increasing; then from the nature of the series, t = atrd;
(1) t' =l-rd;
(2) whence, by addition,
t+t' = atl. 351. The sum of the terms of an Arithmetical Progression is qual to one half the sum of the two extremes, multiplied by the num bpp of terms. Represent the sum of the series by S; then we have
S -- a+(a+d)+(a+20)+.... tl. By writing the series in the reverse order, we have also S=l+(.md)+(1-2d)+. .ta.
Therefore, by addition,
2S = (a +1)+(1+1)+(a+1)+....+(a+1). Now equation (3) expresses the sum of n terms, each equal to (a+1); hence,
2S = n(a+1); and dividing by 2, we obtain the formula, ·
(B) 2 352. To insert any number of arithmetical means between two given terms.
Let n' denote the number of means to be inserted. Then the number of terms in the completed series will be n'+2; and we shall have
n = n'+2. This value of n substituted in formula (A), (349), gives
(0) Having the common difference, the means are readily obtained.
APPLICATION OF THE FORMULAS.
353. The two formulas,
(B) contain in all five quantities, a, l, n, d, S, four of which enter each equation. Now if any three of these quantities be given, the other two may be found; for, if the values of the three given quantities be substituted in the formulas, there will result two equations containing only two unknown quantities.
1. The first term of an arithmetical series is 5, the common difference 3, and the number of terms 24. Find the last term, and the sum of the series.
We have given, a = 5, d=3, n=24; hence, by formula (A), l=5+(24—1)3= 74
Ans. and by formula (B), S= 4(5+74) =948 S
2. Given a =
15, d=-2, and S = 60, to find the number of terms. Substituting the given values in (A) and (B), we have
l=15—2(n-1), 60= (15+);
(2) whence, from (1),
120-15n and from (2),
120-15n = 17n-2n',
n=10 or 6. Both values of n are possible ; for there are two series answering to the given conditions, one having 6 terms, and the other 10;
15, 13, 11, 9, 7, 5, 3, 1, -1, -3, and 15, 13, 11, 9, 7, 5.
The sum of either series is 60.
EXAMPLES FOR PRACTICE.
1. The first term of an arithmetical series is 7, the comnion difference 3, and the number of terms 36; find the last term.
Ans. 112. 2. The first term of an arithmetical series is 275, the last term 5, and the number of terms 46; required the sum of the terms.
Ans. 6440. 3. The sum of an arithmetical series is 156, the number of terms 8, and the common difference 5. Required the two extremes.
4. Find the sum of the terms in an arithmetical progression, knowing that the first term is 1, the common difference d, and the number of terms 101.