The Binomial Formula and Binomial Coefficients

Factorial $n$

If $n=1,2,3,\dots$ factorial $n$ or $n$ factorial is defined as

1. $n! =1\cdot 2\cdot 3\cdot \cdots n$

We also define $zero factorial$ as

2. $0!=1$

Binomial Formula For Positive Integral $n$ 

If $n=1,2,3,\dots$ then

3. $(x+y)^n=x^n+nx^{n-1}y+\cfrac{n(n-1)}{2!}x^{n-2}y^2+\cfrac{n(n-1)(n-2)}{3!}x^{n-3}y^3+\dots +y^n$

This is called the $binomial formula$ . It can be extended to other values of $n$ and then is an infinite series 

Binomial Coefficients 

The result 3 can also be written

4. $(x+y)^n =x^n+\binom{n}{1}x^{n-1}y+\binom{n}{2}x^{n-2}y^2+\binom{n}{3}x^{n-3}y^3+\dots +\binom{n}{n}y^n$

where the coefficients ,called $binomial coefficients$ are given by

5. $\binom{n}{k} =\cfrac{n(n-1)(n-2)\cdots (n-k+1)}{k!}=\cfrac{n!}{k!(n-k)!}=\binom{n}{n-k}$

Properties Of Binomial Coefficients 

6. $\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}$

This leads to Pascal's triangle 

7. $\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots +\binom{n}{n}=2^n$

8. $\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\cdots (-1)^n\binom{n}{n}=0$

9. $\binom{n}{n}+\binom{n+1}{n}+\binom{n+2}{n}+\cdots +\binom{n+m}{n}=\binom{n+m+1}{n+1}$

10. $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots =2^{n-1}$

11. $\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\cdots =2^{n-1}$

12. $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots +\binom{n}{n}^2=\binom{2n}{n}$

13. $\binom{m}{0}\binom{n}{p}+\binom{m}{1}\binom{n}{p-1}+\cdots +\binom{m}{p}\binom{n}{0}=\binom{m+n}{p}$

14. $(1)\binom{n}{1}+(2)\binom{n}{2}+(3)\binom{n}{3}+\cdots +(n)\binom{n}{n}=n2^{n-1}$

15. $(1)\binom{n}{1}-(2)\binom{n}{2}+(3)\binom{n}{3}-\cdots (-1)^{n+1}(n)\binom{n}{n}=0$

Multinomial  Formula

16. $(x_1+x_2+\cdots +x_p)^n=\sum\cfrac{n!}{n_1!n_2!\cdots n_p!}x_1^{n_1}x_2^{n_2}\cdots x_p^{n_p}$

where the sum, denoted by $\sum$ is taken over all nonnegative integers $n_1,n_2, \dots n_p$ for which $n_1+n_2+\cdots +n_p =n$