Question

Using Pascal triangle, write down the binomial expansion of


`{(1+y)}^(8)`
simplifying all the terms.​

Answer 1

1 + 8y + 28y² + 56y³ + 70
`y^(4)` + 56
`y^(5)` + 28
`y^(6)` + 8
`y^(7)` +
`y^(8)`

Explanation:

For the expansion of
`(1+y)^(8)`

Using the row of Pascal's triangle for n = 8 , that is the coefficients are

1 8 28 56 70 56 28 8 1

Decreasing powers of 1 from
`1^(8)` to
`1^(0)`

Increasing powers of y from
`y^(0)` to
`y^(8)`

Then

`(1+y)^(8)`

= 1 .
`1^(8)`.
`y^(0)` + 8.
`1^(7)`.
`y^(1)` + 28.
`1^(6)`.y² + 56.
`1^(5)`.y³ + 70.
`1^(4)`.
`y^(4)` + 56. 1³.
`y^(5)` + 28. 1².
`y^(6)` + 8.
`1^(1)`.
`y^(7)` + 1.
`1^(0)`.
`y^(8)`

= 1 + 8y + 28y² + 56y³ + 70
`y^(4)` + 56
`y^(5)` + 28
`y^(6)` + 8
`y^(7)` +
`y^(8)`

Answer 2

• y⁸ + 8y⁷ + 28y⁶ + 56y⁵ + 70y⁴ + 56y³ + 28y² + 8y + 1

Explanation:

Given binomial

• (1 + y)⁸

Expanding using Pascal triangle (attached)

Replace a and b with y and 1 in the bottom row of the triangle

• (1 + y)⁸ =
• (y + 1)⁸ =
• y⁸ + 8y⁷ + 28y⁶ + 56y⁵ + 70y⁴ + 56y³ + 28y² + 8y + 1