Key Equations

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} \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	$a_{n} = a_{n - 1} + d, n \geq 2 a_{n} = a_{n - 1} + d, n \geq 2$
explicit formula for nth term of an arithmetic sequence	$\begin{array}{l} a_{n} = a_{1} + d (n - 1) \end{array} \begin{array}{l} a_{n} = a_{1} + d (n - 1) \end{array}$

recursive formula for $n t h n t h$ term of a geometric sequence	$a_{n} = r a_{n - 1}, n \geq 2 a_{n} = r a_{n - 1}, n \geq 2$
explicit formula for $n t h n t h$ term of a geometric sequence	$a_{n} = a_{1} r^{n - 1} a_{n} = a_{1} r^{n - 1}$

sum of the first $n n$ terms of an arithmetic series	$S_{n} = \frac{n (a_{1} + a_{n})}{2} S_{n} = \frac{n (a_{1} + a_{n})}{2}$
sum of the first $n n$ terms of a geometric series	$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1 S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1$
sum of an infinite geometric series with $- 1 < r < 1 - 1 < r <$	$S_{n} = \frac{a_{1}}{1 - r}, r \neq 1 S_{n} = \frac{a_{1}}{1 - r}, r \neq 1$

number of permutations of $n n$ distinct objects taken $r r$ at a time	$P (n, r) = \frac{n!}{(n - r)!} P (n, r) = \frac{n!}{(n - r)!}$
number of combinations of $n n$ distinct objects taken $r r$ at a time	$C (n, r) = \frac{n!}{r! (n - r)!} C (n, r) = \frac{n!}{r! (n - r)!}$
number of permutations of $n n$ non-distinct objects	$\frac{n!}{r_{1}! r_{2}! \dots r_{k}!} \frac{n!}{r_{1}! r_{2}! \dots r_{k}!}$

Binomial Theorem	${(x + y)}^{n} = \sum_{k - 0}^{n} (\begin{matrix} n \\ k \end{matrix}) x^{n - k} y^{k} {(x + y)}^{n} = \sum_{k - 0}^{n} (\begin{matrix} n \\ k \end{matrix}) x^{n - k} y^{k}$
$(r + 1) t h (r + 1) t h$ term of a binomial expansion	$(\begin{matrix} n \\ r \end{matrix}) x^{n - r} y^{r} (\begin{matrix} n \\ r \end{matrix}) x^{n - r} y^{r}$

probability of an event with equally likely outcomes	$P (E) = \frac{n (E)}{n (S)} P (E) = \frac{n (E)}{n (S)}$
probability of the union of two events	$P (E \cup F) = P (E) + P (F) - P (E \cap F) P (E \cup F) = P (E) + P (F) - P (E \cap F)$
probability of the union of mutually exclusive events	$P (E \cup F) = P (E) + P (F) P (E \cup F) = P (E) + P (F)$
probability of the complement of an event	$P (E') = 1 - P (E) P (E') = 1 - P (E)$