Expand Using Pascal's Triangle (x+y)^8

${(x + y)}^{8}$

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

$1 - 6 - 15 - 20 - 15 - 6 - 1$

$1 - 7 - 21 - 35 - 35 - 21 - 7 - 1$

$1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(x + y)}^{8}$ , $n = 8$ so the coefficients of the expansion will correspond with line $9$ .

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 8 - 28 - 56 - 70 - 56 - 28 - 8 - 1$ .

$1 a^{8} b^{0} + 8 a^{7} b + 28 a^{6} b^{2} + 56 a^{5} b^{3} + 70 a^{4} b^{4} + 56 a^{3} b^{5} + 28 a^{2} b^{6} + 8 a b^{7} + 1 a^{0} b^{8}$

Substitute the actual values of $a$ $x$ and $b$ $y$ into the expression.

$1 {(x)}^{8} {(y)}^{0} + 8 {(x)}^{7} {(y)}^{1} + 28 {(x)}^{6} {(y)}^{2} + 56 {(x)}^{5} {(y)}^{3} + 70 {(x)}^{4} {(y)}^{4} + 56 {(x)}^{3} {(y)}^{5} + 28 {(x)}^{2} {(y)}^{6} + 8 {(x)}^{1} {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Simplify each term.

Multiply ${(x)}^{8}$ by $1$ .

${(x)}^{8} {(y)}^{0} + 8 {(x)}^{7} {(y)}^{1} + 28 {(x)}^{6} {(y)}^{2} + 56 {(x)}^{5} {(y)}^{3} + 70 {(x)}^{4} {(y)}^{4} + 56 {(x)}^{3} {(y)}^{5} + 28 {(x)}^{2} {(y)}^{6} + 8 {(x)}^{1} {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Anything raised to $0$ is $1$ .

$x^{8} \cdot 1 + 8 {(x)}^{7} {(y)}^{1} + 28 {(x)}^{6} {(y)}^{2} + 56 {(x)}^{5} {(y)}^{3} + 70 {(x)}^{4} {(y)}^{4} + 56 {(x)}^{3} {(y)}^{5} + 28 {(x)}^{2} {(y)}^{6} + 8 {(x)}^{1} {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Multiply $x^{8}$ by $1$ .

$x^{8} + 8 {(x)}^{7} {(y)}^{1} + 28 {(x)}^{6} {(y)}^{2} + 56 {(x)}^{5} {(y)}^{3} + 70 {(x)}^{4} {(y)}^{4} + 56 {(x)}^{3} {(y)}^{5} + 28 {(x)}^{2} {(y)}^{6} + 8 {(x)}^{1} {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Simplify.

$x^{8} + 8 x^{7} y + 28 {(x)}^{6} {(y)}^{2} + 56 {(x)}^{5} {(y)}^{3} + 70 {(x)}^{4} {(y)}^{4} + 56 {(x)}^{3} {(y)}^{5} + 28 {(x)}^{2} {(y)}^{6} + 8 {(x)}^{1} {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Simplify.

$x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x {(y)}^{7} + 1 {(x)}^{0} {(y)}^{8}$

Multiply ${(x)}^{0}$ by $1$ .

$x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + {(x)}^{0} {(y)}^{8}$

Anything raised to $0$ is $1$ .

$x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + 1 {(y)}^{8}$

Multiply ${(y)}^{8}$ by $1$ .

$x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + y^{8}$

Expand Using Pascal's Triangle (x+y)^8

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