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.