Formulas and Results of Permutations Combinations and Binomial Theorem

1. Factorial of n is defined as n! = n \cdot (n - 1) \cdot (n -  2) \cdot (n - 3) \ldots 3 \cdot 2 \cdot 1, where is 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 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 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.

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