Expand Using Pascal's Triangle (r+s)^6

${(r + s)}^{6}$

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 ${(r + s)}^{6}$ , $n = 6$ 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 $a$ $r$ and $b$ $s$ into the expression.

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

Step 4

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Simplify.

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

Simplify.

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

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

$r^{6} + 6 r^{5} s + 15 r^{4} s^{2} + 20 r^{3} s^{3} + 15 r^{2} s^{4} + 6 r s^{5} + {(r)}^{0} {(s)}^{6}$

Anything raised to $0$ is $1$ .

$r^{6} + 6 r^{5} s + 15 r^{4} s^{2} + 20 r^{3} s^{3} + 15 r^{2} s^{4} + 6 r s^{5} + 1 {(s)}^{6}$

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

$r^{6} + 6 r^{5} s + 15 r^{4} s^{2} + 20 r^{3} s^{3} + 15 r^{2} s^{4} + 6 r s^{5} + s^{6}$

Expand Using Pascal's Triangle (r+s)^6

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