The disadvantage of games of chance, the advantage of not exposing to the same danger the whole benefit that is expected, and all the similar results indicated by common sense, subsist, whatever may be the function of the physical fortune which for each individual expresses his moral fortune. It is enough that the proportion of the increase of this function to the increase of the physical fortune diminishes in the measure that the latter increases.
_CONCERNING THE ANALYTICAL METHODS OF THE CALCULUS OF PROBABILITIES._
The application of the principle which we have just expounded to the various questions of probability requires methods whose investigation has given birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences.
If we form the product of the binomials, unity plus the first letter, unity plus the second letter, unity plus the third letter, and so on up to _n_ letters, and subtract unity from this developed product, the result will be the sum of the combination of all these letters taken one by one, two by two, three by three, etc., each combination having unity for a coefficient. In order to have the number of combinations of these _n_ letters taken _s_ by _s_ times, we shall observe that if we suppose these letters equal among themselves, the preceding product will become the _n_th power of the binomial one plus the first letter; thus the number of combinations of _n_ letters taken _s_ by _s_ times will be the coefficient of the _s_th power of the first letter in the 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.