Formula for a factorial

\begin{array}{l} 0! = 1 \\ 1! = 1 \\ n! = n (n - 1) (n - 2) \dots (2) (1), for n \geq 2 \end{array}

recursive formula for nth term of an arithmetic sequence

a_{n} = a_{n - 1} + d, n \geq 2

explicit formula for nth term of an arithmetic sequence

\begin{array}{l} a_{n} = a_{1} + d (n - 1) \end{array}

recursive formula for

n t h

term of a geometric sequence

a_{n} = r a_{n - 1}, n \geq 2

explicit formula for

n t h

term of a geometric sequence

a_{n} = a_{1} r^{n - 1}

sum of the first

n

terms of an arithmetic series

S_{n} = \frac{n (a_{1} + a_{n})}{2}

sum of the first

n

terms of a geometric series

S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1

sum of an infinite geometric series with

- 1 < r < 1

S_{n} = \frac{a_{1}}{1 - r}, r \neq 1

number of permutations of

n

distinct objects taken

r

at a time

P (n, r) = \frac{n!}{(n - r)!}

number of combinations of

n

distinct objects taken

r

at a time

C (n, r) = \frac{n!}{r! (n - r)!}

number of permutations of

n

non-distinct objects

\frac{n!}{r_{1}! r_{2}! \dots r_{k}!}

Binomial Theorem

{(x + y)}^{n} = \sum_{k - 0}^{n} (\begin{matrix} n \\ k \end{matrix}) x^{n - k} y^{k}

(r + 1) t h

term of a binomial expansion

(\begin{matrix} n \\ r \end{matrix}) x^{n - r} y^{r}

probability of an event with equally likely outcomes

P (E) = \frac{n (E)}{n (S)}

probability of the union of two events

P (E \cup F) = P (E) + P (F) - P (E \cap F)

probability of the union of mutually exclusive events

P (E \cup F) = P (E) + P (F)

probability of the complement of an event

P (E') = 1 - P (E)