Question

In the following exercises, expand each binomial using Pascal's Triangle.

$(m+n{)}^{10}$

Short answer

The expansion is$m^{10}+10m^{9}n+45m^8{n}^2+120m^7{n}^3+210m^6{n}^4+252m^5{n}^5+210m^4{n}^6+120m^3{n}^7+45m^2{n}^8+10m{n}^9+n^{10}$

Step-by-step solution

Given Information

Expand binomial using Pascal's Triangle

$(m+n{)}^{10}$

Solution

We have

The exponent is 10 hence eleventh row need to be considered

The expansion is

$\begin{array}{rcl}(m+n{)}^{10}& =& 1m^{10}+10m^{9}n+45m^8{n}^2+120m^7{n}^3+210m^6{n}^4+252m^5{n}^5+210m^4{n}^6+120m^3{n}^7+45m^2{n}^8+10m{n}^9+1n^{10}\\ & =& m^{10}+10m^{9}n+45m^8{n}^2+120m^7{n}^3+210m^6{n}^4+252m^5{n}^5+210m^4{n}^6+120m^3{n}^7+45m^2{n}^8+10m{n}^9+n^{10}\end{array}$