Expand Using Pascal's Triangle (x+y)^12

${(x + y)}^{12}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

$1 - 11 - 55 - 165 - 330 - 462 - 462 - 330 - 165 - 55 - 11 - 1$

$1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1$

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 ${(x + y)}^{12}$ , $n = 12$ so the coefficients of the expansion will correspond with line $13$ .

Step 2

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 $1 - 12 - 66 - 220 - 495 - 792 - 924 - 792 - 495 - 220 - 66 - 12 - 1$ .

$1 a^{12} b^{0} + 12 a^{11} b + 66 a^{10} b^{2} + 220 a^{9} b^{3} + 495 a^{8} b^{4} + 792 a^{7} b^{5} + 924 a^{6} b^{6} + 792 a^{5} b^{7} + 495 a^{4} b^{8} + 220 a^{3} b^{9} + 66 a^{2} b^{10} + 12 a b^{11} + 1 a^{0} b^{12}$

Step 3

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

$1 {(x)}^{12} {(y)}^{0} + 12 {(x)}^{11} {(y)}^{1} + 66 {(x)}^{10} {(y)}^{2} + 220 {(x)}^{9} {(y)}^{3} + 495 {(x)}^{8} {(y)}^{4} + 792 {(x)}^{7} {(y)}^{5} + 924 {(x)}^{6} {(y)}^{6} + 792 {(x)}^{5} {(y)}^{7} + 495 {(x)}^{4} {(y)}^{8} + 220 {(x)}^{3} {(y)}^{9} + 66 {(x)}^{2} {(y)}^{10} + 12 {(x)}^{1} {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

Step 4

Simplify each term.

Multiply ${(x)}^{12}$ by $1$ .

${(x)}^{12} {(y)}^{0} + 12 {(x)}^{11} {(y)}^{1} + 66 {(x)}^{10} {(y)}^{2} + 220 {(x)}^{9} {(y)}^{3} + 495 {(x)}^{8} {(y)}^{4} + 792 {(x)}^{7} {(y)}^{5} + 924 {(x)}^{6} {(y)}^{6} + 792 {(x)}^{5} {(y)}^{7} + 495 {(x)}^{4} {(y)}^{8} + 220 {(x)}^{3} {(y)}^{9} + 66 {(x)}^{2} {(y)}^{10} + 12 {(x)}^{1} {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

Anything raised to $0$ is $1$ .

$x^{12} \cdot 1 + 12 {(x)}^{11} {(y)}^{1} + 66 {(x)}^{10} {(y)}^{2} + 220 {(x)}^{9} {(y)}^{3} + 495 {(x)}^{8} {(y)}^{4} + 792 {(x)}^{7} {(y)}^{5} + 924 {(x)}^{6} {(y)}^{6} + 792 {(x)}^{5} {(y)}^{7} + 495 {(x)}^{4} {(y)}^{8} + 220 {(x)}^{3} {(y)}^{9} + 66 {(x)}^{2} {(y)}^{10} + 12 {(x)}^{1} {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

Multiply $x^{12}$ by $1$ .

$x^{12} + 12 {(x)}^{11} {(y)}^{1} + 66 {(x)}^{10} {(y)}^{2} + 220 {(x)}^{9} {(y)}^{3} + 495 {(x)}^{8} {(y)}^{4} + 792 {(x)}^{7} {(y)}^{5} + 924 {(x)}^{6} {(y)}^{6} + 792 {(x)}^{5} {(y)}^{7} + 495 {(x)}^{4} {(y)}^{8} + 220 {(x)}^{3} {(y)}^{9} + 66 {(x)}^{2} {(y)}^{10} + 12 {(x)}^{1} {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

Simplify.

$x^{12} + 12 x^{11} y + 66 {(x)}^{10} {(y)}^{2} + 220 {(x)}^{9} {(y)}^{3} + 495 {(x)}^{8} {(y)}^{4} + 792 {(x)}^{7} {(y)}^{5} + 924 {(x)}^{6} {(y)}^{6} + 792 {(x)}^{5} {(y)}^{7} + 495 {(x)}^{4} {(y)}^{8} + 220 {(x)}^{3} {(y)}^{9} + 66 {(x)}^{2} {(y)}^{10} + 12 {(x)}^{1} {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

Simplify.

$x^{12} + 12 x^{11} y + 66 x^{10} y^{2} + 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x {(y)}^{11} + 1 {(x)}^{0} {(y)}^{12}$

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

$x^{12} + 12 x^{11} y + 66 x^{10} y^{2} + 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x y^{11} + {(x)}^{0} {(y)}^{12}$

Anything raised to $0$ is $1$ .

Multiply ${(y)}^{12}$ by $1$ .

Expand Using Pascal's Triangle (x+y)^12

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