Modern Math : Permutation and Combination, and, Probability

Multiply the number of ways to do a piece of work with another number of ways to do the work if the works are mutually inclusive or related. Search for the usual keyword: AND defining the statement. 

Add the number of ways to do a piece of work with another number of ways to do the work if the works are mutually exclusive or independent. Search for the usual keyword: AND defining the statement.

Permutation: Each of several possible ways in which a set of things can be ordered or arranged.

nPr = (n!)/ (n-r)!

For n things taken r at a time, permutations = nPr.

For n things taken all at a time, permutations = nPn or n!

For n things taken r at a time when one item is always there, permutations = r*[(n-1) P(r-1).

For n things taken r at a time when one item never occurs, permutations = (n-1) Pr.

For n objects, circular permutation = n!/n = (n-1)!

Combination: A joining or merging of different parts or qualities in which the component elements are individually distinct.

nCr = [n!]/[(n-r)!*r!]

nCr = nPr / r!

nC0 = nCn = 1

nCr = nC(n-r)

nC(r-1) + nCr = (n+1)Cr.

aCa + (a+1)Ca + …. + bCa = (b+1)C(a+1) , a<=b

(a+b)^n = (sum of){(nCr)*a^r*b^(n-r)}, where r = [0, n].

So, 2^n = (1+1)^n = nC0 + nC1 + nC2 + ……. + nCn.

 

Combination of n things where: 

r is taken at a time and s always occur = (n-a) C(r-a).

r is taken at a time and s never occurs = (n-a)Cr.

Selecting r things when all n items are identical = 1.

Selecting one or more than one things when all n items are identical = (n+1).

Selecting one or more than one things from n items that are distinct
= nC0 + nC1 + … + nCn = 2^n.

 

Ways of dividing (m+n) distinct things into m and n things is
= (m+n)Cn or (m+n)Cm = (m+n)!/(m!*n!).

Ways of dividing (m+n+p) distinct things into m and n things is
= (m+n+p)!/(m!*n!*p!).

Probability = P (event) = (Number of ways favorable for the event)/(Total possibilities).

If probability of occurrence of an event E is and probability of non-occurrence is E’,
odds in favor of event = E:E’.
Odds against the event = E’:E.
E+E’ = 1.