Question

Use Pascal’s Triangle to expand${(x+y)}^5$

Short answer

$x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^2+5x{y}^4+y^5$

Step-by-step solution

. Given Information

We are given an expression$={(x+y)}^5$to solve it using the Pascal's triangle

. Pascal Triangle

The pascal triangle is :

We use this triangle to write the co-efficent of the expression and we explained the expression by using the formula :${(a+b)}^n=a^n+\_\_\_a^{n-1}{b}^1+\_\_\_a^{n-2}{b}^2+...+\_\_\_a^1{b}^{n-1}+b^n$

. Expanding the expression 

We expand the expression and we get :

${(x+y)}^5=\_\_x^5{y}^0+\_\_x^{5-1}{y}^1+\_\_x^{5-2}{y}^2+\_\_x^{5-3}{y}^3+\_\_x^{5-4}{y}^4+\_\_x^{5-5}{y}^5$

${(x+y)}^5=\_\_x^5+\_\_x^{4}y+\_\_x^3{y}^2+\_\_x^2{y}^3+\_\_x{y}^4+\_\_y^5$

. Adding the co-efficient from the Pascal's triangle

After expanding the expression we see that we get $6$expression so we will take the sixth row's co-efficient of the pascal triangle. So we get :

$x^5+5x^{4}y+10x^3{y}^2+10x^2{y}^2+5x{y}^4+y^5$