Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/37

Rh development in this binomial; and this number is obtained by means of the known binomial formula.

Attention must be paid to the respective situations of the letters in each combination, observing that if a second letter is joined to the first it may be placed in the first or second position which gives two combinations. If we join to these combinations a third letter, we can give it in each combination the first, the second, and the third rank which forms three combinations relative to each of the two others, in all six combinations. From this it is easy to conclude that the number of arrangements of which $$s$$ letters are susceptible is the product of the numbers from unity to $$s$$. In order to pay regard to the respective positions of the letters it is necessary then to multiply by this product the number of combinations of $$n$$ letters $$s$$ by $$s$$ times, which is tantamount to taking away the denominator of the coefficient of the binomial which expresses this number.

Let us imagine a lottery composed of $$n$$ numbers, of which $$r$$ are drawn at each draw. The probability is demanded of the drawing of $$s$$ given numbers in one draw. To arrive at this let us form a fraction whose denominator will be the number of all the cases possible or of the combinations of $$n$$ letters taken $$r$$ by $$r$$ times, and whose numerator will be the number of all the combinations which contain the given $$s$$ numbers. This last number is evidently that of the combinations of the other numbers taken $$n$$ less $$s$$ by $$n$$ less $$s$$ times. This fraction will be the required probability, and we shall easily find that it can be reduced to a fraction whose numerator is the number of combinations of $$r$$