<s>[INST] How did Sir Isaac Newton develop and formulate the famous binomial theorem? After completing combination, I have started to read Binomial Theorem. My book only mentioned about Pascal's Triangle. And the formula was then given straightforward. But how did Sir Issac Newton actually develop this theorem?? How did he formulate all these??? Please help. 
 [/INST] About the Binomial Theorem.
When expanding the powers of a binomial by hand and grouping the terms by identical powers, it is not very hard to observe the pattern:
$$(x+y)^0=1$$
$$(x+y)^1=x+y$$
$$(x+y)^2=x^2+2xy+y^2$$
$$(x+y)^3=x^3+3x^2y+3xy^2+y^3$$
$$(x+y)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4$$
$$\dots$$


*

*the expansion is the sum of all $n+1$ possible products of the form $x^iy^j$, with $i+j=n$.

*the shape of Pascal's triangle clearly appears as the number of terms goes linearly increasing.

*the basic recurrence relation between Pascal's numbers appears with the following expansion:


$$\begin{align}(x+y)(x+y)^3
&=x^4+3x^3y+3x^2y^2+xy^3\ +\\
&\ \ \ \ \ \underline{\ \ \ \ +\ \ x^3y+3x^3y^2+3xy^3+y^4}\\
&=x^4+4x^3y+6x^2y^2+4xy^3+y^4
\end{align}$$
You get a new line of coefficients by adding the last line to itself shifted by one position, so that $C_m^{n+1}=C_{m-1}^n+C_m^n$.
All of this is rather straightforward to establish. More interesting is the link with combinatorial analysis.
If you expand $(x+y)^n$ without regrouping the terms,
$$(x+y)^0=1$$
$$\begin{align}(x+y)^1&=x\\
&+y\end{align}$$
$$\begin{align}(x+y)^2&=xx\\
&+xy+yx\\
&+yy\end{align}$$
$$\begin{align}(x+y)^3&=xxx\\
&+xxy+xyx+yxx\\
&+xyy+yxy+yyx\\
&+yyy\end{align}$$
$$\begin{align}(x+y)^4&=xxxx\\
&+xxxy+xxyx+xyxx+yxxx\\
&+xxyy+xyxy+xyyx+yxxy+yxyx+yyxx\\
&+yyxy+yyyx+xyyy+yxyy\\
&+yyyy\end{align}$$
$$\dots$$
you see that there are $2^n$ terms, and you observe that all combinations of $i$ letters $x$ and $j$ letters $y$ (with $i+j=n$) can be grouped as $x^iy^j$, hence
$$C_m^n=\binom nm.$$
At the same time, you see a bell-shaped histogram appear, that prefigures the Gaussian curve.
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