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_ numbers taken _s_ by _s_ times, and whose denominator is the number of combinations of _n_ numbers taken similarly _s_ by _s_ times. Thus in the lottery of France, formed as is known of 90 numbers of which five are drawn at each draw, the probability of drawing a given combination is 5/90, or 1/18; the lottery ought then for the equality of the play to give eighteen times the stake. The total number of combinations two by two of the 90 numbers is 4005, and that of the combinations two by two of 5 numbers is 10. The probability of the drawing of a given pair is then 1/4005, and the lottery ought to give four hundred and a half times the stake; it ought to give 11748 times for a given tray, 511038 times for a quaternary, and 43949268 times for a quint. The lottery is far from giving the player these advantages.