3. To the saine base add the product of the exponent and common difference of the factorial factor thus completed, and to the sum as a base apply the remaining exponent of the two factorial factors, and the common difference, then the quantity thus formed will be the other factorial factor. Examples. 1. Resolve mic into two factorial factors, so that one of them may have the given exponent r. By rule n-r will be the exponent of the other, now if mr be the one factorial, (m+rc) n―rk will be the other. Or if mi be the one factorial [m+(n−r)c]rk will be the other. 2. Resolve male into two factorial factors, so that one of them may have the given exponent 1. By rule n-1 will be the exponent of the other; now, therefore, if m=m, be the one factorial factor, then (m-c) will be the other, or if m- be the one factorial, then will [m+(n-1)c] m+ (n-1)c be the other. 3. Resolve mall into two factorial factors, so that one of them may have the given exponent 1. By rule n-1 will be the exponent of the other; if therefore milr be the one factorial factor, then will (m+1)- be the other, or if the one factorial factor be m"- then will the other factorial factor be (m+n-1)=m+n—1. 4. Resolve male into two factorial factors, so that one of them may have the given exponent r. By rule n―r will be the exponent of the other; therefore, if m be the one factorial factor, then will (m-rc)"-r be the other. Or if mark be the one factorial factor, then will [m-(nr) c]m be the other. 5. Resolve ma into two factorial factors, so that one of them may have the given exponent 1. By rule n-1 is the exponent of the other; therefore if mш=m de the one factorial factor, the other will be (m-1)-III. Or, if the one factorial factor be ma- the other will be (m—n+1)Ã=m—n+1, which is the last term of the factorial mal Definitions. 1. The different positions in which the same things can be placed, are called their permutations. 2. The different collections that can be formed out of a given number of things, are called combinations. 3. Each collection of things combined out of a given number of things, is called a parcel. 4. The number of things to be combined, are called the base or first order, and the things combined are said to be raised from that base. 5. The whole number of parcels that can be formed by taking a given number of things, out of a greater given number of things into each parcel, is called an order of combination. 6. The number of an order of combination, is the same as the number of the letters forming each parcel. 7. Combinations which have no two letters alike in any parcel, are called combinations without repetitions. 8. Combinations which have parcels containing two or more things alike, are called combinations with repetitions. Theorem 1. (1.) The number of permutations of m things taken n at a time is m As any one of the m quantities may stand first, there will be m different ways of filling up the first place. Now whichever of the m things stand first, there will remain (m-1) things to make choice of to fill up the second place; let, therefore, each of the m things be supposed to be placed and to stand first when combined in separate parcels; then since any one of them may be adjoined with the remaining m— -1 things m-1 different ways, let this be done, and each one of the m things will be multiplied m-1 times when placed in two's; therefore the whole number of the m things will be multiplied m-1 times; and therefore the number of parcels in two's will be m(m-1)=m2 which is equal to the number of their permutations. Now, since each parcel contains two things, there will remain m −2 things; therefore each parcel in two's may be adjoined m-2 times with each of the remaining things; let this be done, and each parcel in two's will be multiplied m-2 times; and consequently the whole number of parcels in two's will be multiplied M- -2 times, in order to produce the parcels in three's. But the number of parcels in two's is m(m-1); therefore the number of parcels in three's is m(m-1) (m-2)=m31 In the same manner it may be shown that the number of parcels or permutations in four's is m4; therefore, in general, the number of parcels or permutations with m things, taken n at a time, is m = (2.) Corollary:-When n is equal to m, the number of permutations of things, taken m at a time, is mm1; hence the permutations of two things in two's is 1x2=2; of three things in three's, is 1x2 x3=6; and of four things in four's, is 1 x 2×3×4= 24; and so on. Theorem 2. (3.) The number of combinations, without repetitions, raised from a base taken m things, taken n things at a time, is The number of permutions for m things in two's is m2; but the permutations of two things is 1.2=2; therefore two things has 1.2=2 more permutations than combinations, and consequently in the whole number mal of permutations there are 1.2 times more permutations than combinations; therefore the number of combinations of mar 1211° m things, taken two at a time, are In the same manner, as every parcel of three's admits of 1.2.3 1311=6 more permutations than combinations, but the permutations of ➡ things in three's is m3; therefore the number of combinations of 113317 things in three's is In general since the number of permutations of m things taken n at a time is m, and the number of permutations of n things taken " at a time is 1", the number of combinations raised from a base of (4.) The number of parcels, with n things in each, having repe titions raised from a base of r things, is 1 For the number of parcels with n things in each, having no repe are the first and last terms of the factorial m", therefore (m-n+1) 11: ས་་ lu But the number of things in the base required to raise a class of combinations with n things in each parcel without repetitions, is greater by (n-1) than the number of things in the base required to raise the same number of parcels with the same number of things in each, having repetitions. Therefore, deducting n-1 from m, gives m―n+1 for the number of things in the base (m-n+1)"11 required to raise parcels with n things in each. Now let r-m-n+1; then will be the number of parcels ix with n things in each, raised from a base of r things. ON CHANCES. (1.) If an event may take place in n different ways, and each of these be equally likely to happen, the probability that it will take place in a specified way is properly represented by , certainty being n represented by unity or, which is the same thing, if the value of certainty be unity, the value of the expectation that the event will happen in a specified way is For, the sum of all the probabilities is certainty, or unity, because the event must take place in some one of the ways, and the probabilities are equal; therefore each of them is 1 n (2.) Cor. If the value of certainty be a, the value of the expecBut in the following articles we suppose the value of tation is a n certainty to be unity. (3.) If an event may happen in a ways, and fail in ways any of these being equally probable, the chance of its happening is b The chance of its happening must, from the nature of the supposition, be to the chance of its failing, as a to b; therefore the chance of its happening, together with the chance of its failing aa+b; and the event must either happen or fail; consequently, the chance of its happening, together with the chance of its failing, is certainty; hence, the chance of its happening certainty :: a:+b; and the Also, since the chance of its happening together with the chance of its failing is certainty, which is represented by unity, 1- that is, a a+b (4.) Er. 1. The probability of throwing an ace with a single die, in one trial, is; the probability of not throwing an ace is 6 1 5 2 the probability of throwing either an ace or a deuce, is ; &c. 6 |