binomial shit

binomial expression

an expression in the form of (x + y)^n

binomial coefficient

(x+y)^n = nΣ(r=0) x^(n-r)y^r

nCr

gives the coefficient for the rth term of (kx+c)^n nCr = n!/r!(n-r)!

pascal's triangle

used to help binomial expansion- the (n+1)th row gives the coefficients in the expansion of (a+b)^n

formed by adding adjacent pairs of numbers to find the numbers on the next row

pascal's identity

nCr = (n-1)C(r-1) + (n-1)Cr

r <= n

derangement

a permutation of all of the elements of a finite set that leaves no object in its original position real-world example: students must mark eachothers' papers, no student may mark their own paper, a derangement of the set of students must be made

inclusion-exclusion principle for derangements

!n = n![1 - 1/1! + 1/2! … (-1)^n/n!]

equal to !n = n! - n!/1! + n!/2! … (-1)^n

approximation = n!/e (rounding this approximation to the nearest integer gives the number of derangements)