Question

(a + b)2 = 1a^2 + 2ab + 1b^2

(a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3

How are binomial expansions related to Pascal’s triangle?

Answer 1

The coefficients of the terms come from rows of the triangle.

If the exponent is n, look at the entries in row n.

Answer 2

Consider the binomial
`\displaystyle{ (a+b)^\displaystyle{n`,

where n=0, 1, 2, 3, ...

For example


`\displaystyle{ (a+b)^0=1`


`\displaystyle{(a+b)^1=1a+1b`


`\displaystyle{(a+b)^2=1a^2+2ab+1b^2`


`\displaystyle{ (a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3`
...
...


Consider the Pascal's triangle, as shown in the picture, where the very first row is denoted by row 0, the second by row 1, the third by row 2 and so on...


We notice that the coefficients of the expansion of
`(a+b)^n`, are the entries in the
`n^(th)` row of Pascal's triangle.