The value of a circulating decimal is a fraction whose numerator is the period, and denominator a number consisting of as many nines as there are digits in the period. (W. 225; F. 50-2.) (23.) Any three of the five quantities a, a,, r, n, s, being given, the other two may be found from the following formulæ. (24.) The number of permutations of n things taken r at a time (W. 226-31; E. 352-60; Bour. 146-50.) " The equations containing logarithms, are placed here on account of their connexion with the rest, although the properties of logarithms have not yet been stated. # The number of permutations = [2. The number of combinations = n the m + 1th term being n (n − 1) (n − 2).... (n − m + 1) If n is a positive integer, then the number of terms n + 1; an-mbm the coefficients of terms equally distant from both extremities of the series are equal; the coefficient of every term is an integer; also the theorem may be put under the following forms: The product of n simple factors, on which the binomial theorem depends, may be thus expressed: |