Expand Using Pascal's Triangle (m+n)^6

${(m + n)}^{6}$

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

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

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

Substitute the actual values of $a$ $m$ and $b$ $n$ into the expression.

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

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

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

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

Simplify.

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

Simplify.

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

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

$m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + {(m)}^{0} {(n)}^{6}$

Anything raised to $0$ is $1$ .

$m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + 1 {(n)}^{6}$

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

$m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + n^{6}$

Expand Using Pascal's Triangle (m+n)^6

Related Questions

Support LCMGCF.com by disabling your adblocker.