Formulas and Results of Permutations Combinations and Binomial Theorem

1. The factorial of n is defined as n! = n \cdot (n - 1) \cdot (n - 2) \cdot (n - 3) \ldots 3 \cdot 2 \cdot 1, where there is a natural number.

2. {}^n{P_r} = \frac{{n!}}{{(n - r)!}} (Permutation)

3. {}^n{C_r} = \left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right) = \frac{{n!}}{{r!(n - r)!}} (Combination)

4. \left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right) = \left( {\begin{array}{*{20}{c}}n \\ {n - r} \end{array}} \right)

5. \left( {\begin{array}{*{20}{c}}n \\ r \end{array}} \right) = \left( {\begin{array}{*{20}{c}}n \\ {r + 1} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {n + 1} \\ {r + 1} \end{array}} \right) \forall n,r \in \mathbb{N}

6. {}^n{C_r} = \frac{{{}^n{P_r}}}{{r!}}

7. \left( {\begin{array}{*{20}{c}}2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}}3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \cdots + \left( {\begin{array}{*{20}{c}} {n + 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {n + 2} \\ 3 \end{array}} \right)

8. For large n, n! = \sqrt {2\pi n} {\text{ }}{n^n}{\text{ }}{e^{ - n}} is called the Striling Approximation Formula.

9. \frac{n}{2} = \left( {\begin{array}{*{20}{c}}n \\ 0 \end{array}} \right) + \left( {\begin{array}{*{20}{c}}n \\ 1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) + \cdots + \left( {\begin{array}{*{20}{c}}n \\ n \end{array}} \right)

10. \left( {\begin{array}{*{20}{c}}n \\ 0 \end{array}} \right) = 1, \left( {\begin{array}{*{20}{c}}n \\ 1 \end{array}} \right) = n, \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) = \frac{{n(n - 1)}}{{2!}}, \left( {\begin{array}{*{20}{c}}n \\ 3 \end{array}} \right) = \frac{{n(n - 1)(n - 2)}}{{3!}}

11. \left( {\begin{array}{*{20}{c}}n \\ {n - 1} \end{array}} \right) = n, \left( {\begin{array}{*{20}{c}} n \\ n \end{array}} \right) = 1

12. If n is positive integer, then {(a + b)^n} = \left( {\begin{array}{*{20}{c}}n \\ 0 \end{array}} \right){a^n} + \left( {\begin{array}{*{20}{c}} n \\ 1 \end{array}} \right){a^{n - 1}}b + \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right){a^{n - 2}}{b^2} + \cdots + \left( {\begin{array}{*{20}{c}}n \\ r \end{array}} \right){a^{n - r}}{b^r} + \cdots + \left( {\begin{array}{*{20}{c}}n \\ n \end{array}} \right){b^n}

13. \begin{gathered}{(a + b)^n} = {a^n} + n{a^{n - 1}}b + \frac{{n(n - 1)}}{{2!}}{a^{n - 2}}{b^2} + \cdots \\ \cdots + \frac{{n(n - 1)(n - 2) \cdots (n - r + 1)}}{{r!}}{a^{n - r}}{b^r} + \cdots + {b^n} \\ \end{gathered}

14. General Term = \left( {\begin{array}{*{20}{c}}n \\ r \end{array}} \right){a^{n - r}}{b^r}

15. If n is not a positive integer, then the binomial expansion is {(1 + x)^n} = 1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)(n - 2)}}{{3!}} + \cdots is called the binomial series.