Raising Binomial to the Natural Power (Newton's Binom Formula)

Here we will talk about how to raise `a+b` to any natural power.

if `n=1`, then `(a+b)^1=a+b`.

If `n=2`, then `(a+b)^2=a^2+2ab+b^2`.

If `n=3`, then `(a+b)^3=a^3+3a^2b+3ab^2+b^3`.

Using the fact that `(a+b)^4=(a+b)^3xx(a+b)`, we can write the following formula `(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4`.

In general, the following formula is correct `color(blue)((a+b)^n=a^n+C_n^1a^(n-1)b+C_n^2a^(n-2)b^2+...+C_n^ka^(n-k)b^k+...+C_n^(n-1)ab^(n-1)+b^n)` and is called Newton's binom formula.

`C_n^0, C_n^1=n`, `C_n^2=(n*(n-1))/2` ,...,

`C_n^k=(n(n-1)(n-2)...(n-k+1))/(1*2*3...k)` ,...,

`C_n^(n-1)=n, C_n^n=1` - are binomial coefficients ( the numbers of combinations with the elements of zero, one, two,..., `k` ,...,`n-1`, `n`-elements (see the notes the combinations and their properties and binom of Newton).

For example,

`(a+b)^6=a^6+6a^5b+(6*5)/(1*2)a^4b^2+(6*5*4)/(1*2*3)a^3b^3+(6*5*4*3)/(1*2*3*4)a^2b^4+(6*5*4*3*2)/(1*2*3*4*5)ab^5++b^6= a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6`