User:Saul/algebra

Combinations and Permutations

Permutations/Pick

A permutation is a sequence where the order does matter.
This is often notated by n pick r where n is the data set - say 40 lotto balls and r is the selection count - say 8 balls.
The general formulae for permutations is:
P = n! / (n-k)!

Combinations/Choose

A combination is a sequence where the order does NOT matter.
This is often notated by n choose r - n and r mean the same thing as in pick.
The general formulae for combinations is:
C = n! / ((n-k)!k!)
Also note that:
n choose k is the same as n choose (n - k)

Binomial Theorem

(a + b)ⁿ = _k=0Σⁿ ( ( n choose k ) a^k b^n-k )

Approximation Using Binomial

Calculating a number to the power of another can be approximated by the following formulae:

(1 + x)ⁿ = 1 + nx + (n(n-1)x²)/2! + (n(n-1)(n-2)x³)/3! + (n(n-1)(n-2)(n-3)x⁴)/4! ...

Note: This is unending so the more steps you take the more accurate the number will be, however usually 3-4 steps is enough.