Expand Using Pascal's Triangle (x+y)^10

${(x + y)}^{10}$

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

$1 - 9 - 36 - 84 - 126 - 126 - 84 - 36 - 9 - 1$

$1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$

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 ${(x + y)}^{10}$ , $n = 10$ so the coefficients of the expansion will correspond with line $11$ .

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 $1 - 10 - 45 - 120 - 210 - 252 - 210 - 120 - 45 - 10 - 1$ .

$1 a^{10} b^{0} + 10 a^{9} b + 45 a^{8} b^{2} + 120 a^{7} b^{3} + 210 a^{6} b^{4} + 252 a^{5} b^{5} + 210 a^{4} b^{6} + 120 a^{3} b^{7} + 45 a^{2} b^{8} + 10 a b^{9} + 1 a^{0} b^{10}$

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

$1 {(x)}^{10} {(y)}^{0} + 10 {(x)}^{9} {(y)}^{1} + 45 {(x)}^{8} {(y)}^{2} + 120 {(x)}^{7} {(y)}^{3} + 210 {(x)}^{6} {(y)}^{4} + 252 {(x)}^{5} {(y)}^{5} + 210 {(x)}^{4} {(y)}^{6} + 120 {(x)}^{3} {(y)}^{7} + 45 {(x)}^{2} {(y)}^{8} + 10 {(x)}^{1} {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

Simplify each term.

Multiply ${(x)}^{10}$ by $1$ .

${(x)}^{10} {(y)}^{0} + 10 {(x)}^{9} {(y)}^{1} + 45 {(x)}^{8} {(y)}^{2} + 120 {(x)}^{7} {(y)}^{3} + 210 {(x)}^{6} {(y)}^{4} + 252 {(x)}^{5} {(y)}^{5} + 210 {(x)}^{4} {(y)}^{6} + 120 {(x)}^{3} {(y)}^{7} + 45 {(x)}^{2} {(y)}^{8} + 10 {(x)}^{1} {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

Anything raised to $0$ is $1$ .

$x^{10} \cdot 1 + 10 {(x)}^{9} {(y)}^{1} + 45 {(x)}^{8} {(y)}^{2} + 120 {(x)}^{7} {(y)}^{3} + 210 {(x)}^{6} {(y)}^{4} + 252 {(x)}^{5} {(y)}^{5} + 210 {(x)}^{4} {(y)}^{6} + 120 {(x)}^{3} {(y)}^{7} + 45 {(x)}^{2} {(y)}^{8} + 10 {(x)}^{1} {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

Multiply $x^{10}$ by $1$ .

$x^{10} + 10 {(x)}^{9} {(y)}^{1} + 45 {(x)}^{8} {(y)}^{2} + 120 {(x)}^{7} {(y)}^{3} + 210 {(x)}^{6} {(y)}^{4} + 252 {(x)}^{5} {(y)}^{5} + 210 {(x)}^{4} {(y)}^{6} + 120 {(x)}^{3} {(y)}^{7} + 45 {(x)}^{2} {(y)}^{8} + 10 {(x)}^{1} {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

Simplify.

$x^{10} + 10 x^{9} y + 45 {(x)}^{8} {(y)}^{2} + 120 {(x)}^{7} {(y)}^{3} + 210 {(x)}^{6} {(y)}^{4} + 252 {(x)}^{5} {(y)}^{5} + 210 {(x)}^{4} {(y)}^{6} + 120 {(x)}^{3} {(y)}^{7} + 45 {(x)}^{2} {(y)}^{8} + 10 {(x)}^{1} {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

Simplify.

$x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + 120 x^{7} y^{3} + 210 x^{6} y^{4} + 252 x^{5} y^{5} + 210 x^{4} y^{6} + 120 x^{3} y^{7} + 45 x^{2} y^{8} + 10 x {(y)}^{9} + 1 {(x)}^{0} {(y)}^{10}$

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

$x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + 120 x^{7} y^{3} + 210 x^{6} y^{4} + 252 x^{5} y^{5} + 210 x^{4} y^{6} + 120 x^{3} y^{7} + 45 x^{2} y^{8} + 10 x y^{9} + {(x)}^{0} {(y)}^{10}$

Anything raised to $0$ is $1$ .

$x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + 120 x^{7} y^{3} + 210 x^{6} y^{4} + 252 x^{5} y^{5} + 210 x^{4} y^{6} + 120 x^{3} y^{7} + 45 x^{2} y^{8} + 10 x y^{9} + 1 {(y)}^{10}$

Multiply ${(y)}^{10}$ by $1$ .

$x^{10} + 10 x^{9} y + 45 x^{8} y^{2} + 120 x^{7} y^{3} + 210 x^{6} y^{4} + 252 x^{5} y^{5} + 210 x^{4} y^{6} + 120 x^{3} y^{7} + 45 x^{2} y^{8} + 10 x y^{9} + y^{10}$

Expand Using Pascal's Triangle (x+y)^10

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