Question

What are the coefficients for the binomial expansion of (a + b)^3?

Answer 1

The coefficients are given by
`\dbinom 3k=(3!)/(k!(3-k)!)` where
`k\in\{0,1,2,3\}`. These are, in order,
`\{1,3,3,1\}`.

So the expansion is


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

Answer 2

Required coefficient :coefficient of 1st, 2nd, 3rd and 4th term are 1, 3,3,1 respectively.

Explanation:

Formula for binomial expansion is
`(a+b)^n=^nC_0\cdot a^n\cdot b^0+^nC_1\cdota^(n-1)\cdot b^1+------b^n`

We will substitute the values a=a, b=b and n=3 in the formula for binomial expansion we will get

`^3C_0\cdot a^3\cdot b^0+^3C_1\cdot a^2\cdot b^1+^3C_2\cdot a^1\cdot b^2+^3C_3\cdot a^0\cdot b^3`

After simplification of the terms we will get

`a^3+3\cdot a^2\cdot b+3\cdot a\cdot b^2+b^3`

Since,
`^nC_r=(n!)/((r!)(n-r)!)`

Therefore the required coefficient : coefficients of 1st, 2nd, 3rd and 4th term are 1, 3,3,1 respectively.