Expand Using Pascal's Triangle (v+w)^6

${(v + w)}^{6}$

Step 1

Pascal\'s Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

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

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 ${(v + w)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $v$ and $b$ $w$ into the expression.

$1 {(v)}^{6} {(w)}^{0} + 6 {(v)}^{5} {(w)}^{1} + 15 {(v)}^{4} {(w)}^{2} + 20 {(v)}^{3} {(w)}^{3} + 15 {(v)}^{2} {(w)}^{4} + 6 {(v)}^{1} {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

Step 4

Simplify each term.

Multiply ${(v)}^{6}$ by $1$ .

${(v)}^{6} {(w)}^{0} + 6 {(v)}^{5} {(w)}^{1} + 15 {(v)}^{4} {(w)}^{2} + 20 {(v)}^{3} {(w)}^{3} + 15 {(v)}^{2} {(w)}^{4} + 6 {(v)}^{1} {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

Anything raised to $0$ is $1$ .

$v^{6} \cdot 1 + 6 {(v)}^{5} {(w)}^{1} + 15 {(v)}^{4} {(w)}^{2} + 20 {(v)}^{3} {(w)}^{3} + 15 {(v)}^{2} {(w)}^{4} + 6 {(v)}^{1} {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

Multiply $v^{6}$ by $1$ .

$v^{6} + 6 {(v)}^{5} {(w)}^{1} + 15 {(v)}^{4} {(w)}^{2} + 20 {(v)}^{3} {(w)}^{3} + 15 {(v)}^{2} {(w)}^{4} + 6 {(v)}^{1} {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

Simplify.

$v^{6} + 6 v^{5} w + 15 {(v)}^{4} {(w)}^{2} + 20 {(v)}^{3} {(w)}^{3} + 15 {(v)}^{2} {(w)}^{4} + 6 {(v)}^{1} {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

Simplify.

$v^{6} + 6 v^{5} w + 15 v^{4} w^{2} + 20 v^{3} w^{3} + 15 v^{2} w^{4} + 6 v {(w)}^{5} + 1 {(v)}^{0} {(w)}^{6}$

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

$v^{6} + 6 v^{5} w + 15 v^{4} w^{2} + 20 v^{3} w^{3} + 15 v^{2} w^{4} + 6 v w^{5} + {(v)}^{0} {(w)}^{6}$

Anything raised to $0$ is $1$ .

$v^{6} + 6 v^{5} w + 15 v^{4} w^{2} + 20 v^{3} w^{3} + 15 v^{2} w^{4} + 6 v w^{5} + 1 {(w)}^{6}$

Multiply ${(w)}^{6}$ by $1$ .

$v^{6} + 6 v^{5} w + 15 v^{4} w^{2} + 20 v^{3} w^{3} + 15 v^{2} w^{4} + 6 v w^{5} + w^{6}$

Expand Using Pascal's Triangle (v+w)^6

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