Question

Use Pascal’s Triangle to expand${(p+q)}^7$

Short answer

$q^7+7q^{6}r+21q^5{r}^2+35q^4{r}^3+35q^3{r}^4+21q^2{r}^5+7q{r}^6+r^7$

Step-by-step solution

. Given Information

We are given the expression : ${(p+q)}^7$and we need to solve this using the Pascal's triangle

. Pascal's Triangle

First we need to expand the given expression by using the formula :

${(a+b)}^n=\_\_a^n+\_\_a^{n-1}{b}^1+\_\_a^{n-2}{b}^2+...+\_\_a^1{b}^{n-1}+b^n$

After this we find the co-efficient by using the Pascal's triangle , we count the number of expression obtained and then use that number as the row number of the triangle :

. Expanding the expression 

We need to expand the given expression :

${(p+q)}^7=\_\_p^7+\_\_p^{7-1}q+\_\_p^{7-2}{q}^2+\_\_p^{7-3}{q}^3+\_\_p^{7-4}{q}^4+\_\_p^{7-5}{q}^5+\_\_p^{7-6}{q}^6+\_\_p^{7-7}{q}^7$

${(p+q)}^7=\_\_p^7+\_\_p^{6}q+\_\_p^5{q}^2+\_\_p^4{q}^3+\_\_p^3{q}^4+\_\_p^2{q}^5+\_\_p{q}^6+\_\_q^7$

. Putting the co-efficient

We see that the number of expression obtained after expanding is six so we will take the co-efficient from the sixth row of the Pascal's triangle, so we will get :

$q^7+7q^{6}r+21q^5{r}^2+35q^4{r}^3+35q^3{r}^4+21q^2{r}^5+7q{r}^6+r^7$