Expand Using Pascal's Triangle (3x+y)^11

${(3 x + y)}^{11}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$\dots$

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

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

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

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 ${(3 x + y)}^{11}$ , $n = 11$ so the coefficients of the expansion will correspond with line $12$ .

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

$1 a^{11} b^{0} + 11 a^{10} b + 55 a^{9} b^{2} + 165 a^{8} b^{3} + 330 a^{7} b^{4} + 462 a^{6} b^{5} + 462 a^{5} b^{6} + 330 a^{4} b^{7} + 165 a^{3} b^{8} + 55 a^{2} b^{9} + 11 a b^{10} + 1 a^{0} b^{11}$

Step 3

Substitute the actual values of $a$ $3 x$ and $b$ $y$ into the expression.

$1 {(3 x)}^{11} {(y)}^{0} + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Step 4

Simplify each term.

Multiply ${(3 x)}^{11}$ by $1$ .

${(3 x)}^{11} {(y)}^{0} + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$3^{11} x^{11} {(y)}^{0} + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $11$ .

$177147 x^{11} {(y)}^{0} + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Anything raised to $0$ is $1$ .

$177147 x^{11} \cdot 1 + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $177147$ by $1$ .

$177147 x^{11} + 11 {(3 x)}^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 11 (3^{10} x^{10}) {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $10$ .

$177147 x^{11} + 11 (59049 x^{10}) {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $59049$ by $11$ .

$177147 x^{11} + 649539 x^{10} {(y)}^{1} + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Simplify.

$177147 x^{11} + 649539 x^{10} y + 55 {(3 x)}^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 55 (3^{9} x^{9}) {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $9$ .

$177147 x^{11} + 649539 x^{10} y + 55 (19683 x^{9}) {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $19683$ by $55$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} {(y)}^{2} + 165 {(3 x)}^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 165 (3^{8} x^{8}) {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $8$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 165 (6561 x^{8}) {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $6561$ by $165$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} {(y)}^{3} + 330 {(3 x)}^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 330 (3^{7} x^{7}) {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $7$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 330 (2187 x^{7}) {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $2187$ by $330$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} {(y)}^{4} + 462 {(3 x)}^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 462 (3^{6} x^{6}) {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $6$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 462 (729 x^{6}) {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $729$ by $462$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} {(y)}^{5} + 462 {(3 x)}^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 462 (3^{5} x^{5}) {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $5$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 462 (243 x^{5}) {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $243$ by $462$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} {(y)}^{6} + 330 {(3 x)}^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 330 (3^{4} x^{4}) {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $4$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 330 (81 x^{4}) {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $81$ by $330$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} {(y)}^{7} + 165 {(3 x)}^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 165 (3^{3} x^{3}) {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $3$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 165 (27 x^{3}) {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $27$ by $165$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} {(y)}^{8} + 55 {(3 x)}^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Apply the product rule to $3 x$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} y^{8} + 55 (3^{2} x^{2}) {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Raise $3$ to the power of $2$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} y^{8} + 55 (9 x^{2}) {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $9$ by $55$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} y^{8} + 495 x^{2} {(y)}^{9} + 11 {(3 x)}^{1} {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Simplify.

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} y^{8} + 495 x^{2} y^{9} + 11 (3 x) {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply $3$ by $11$ .

$177147 x^{11} + 649539 x^{10} y + 1082565 x^{9} y^{2} + 1082565 x^{8} y^{3} + 721710 x^{7} y^{4} + 336798 x^{6} y^{5} + 112266 x^{5} y^{6} + 26730 x^{4} y^{7} + 4455 x^{3} y^{8} + 495 x^{2} y^{9} + 33 x {(y)}^{10} + 1 {(3 x)}^{0} {(y)}^{11}$

Multiply ${(3 x)}^{0}$ by $1$ .

Apply the product rule to $3 x$ .

Anything raised to $0$ is $1$ .

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

Anything raised to $0$ is $1$ .

Multiply ${(y)}^{11}$ by $1$ .

Expand Using Pascal's Triangle (3x+y)^11

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