Pascal's Triangle

Rows of Pascal's triangle provide the coefficients to expand $(a+b{)}^n$. Coefficients for $(a+b{)}^n$ can read from the $(n+1{)}^{st}$ row of Pascal's triangle.

For example,

• $(a+b{)}^2=a^2+2ab+b^2$  from the row 1, 2, 1
• $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$  from the row 1, 3, 3, 1
• $(a+b{)}^4=a^4+4a^{3}b+6a^2{b}^2+4a{b}^3+b^4$  from the row 1, 4, 6, 4, 1
• $(a+b{)}^5=a^5+5a^{4}b+10a^3{b}^2+10a^2{b}^3+5a{b}^4+b^5$  from the row 1, 5, 10, 10, 5, 1
• $(a+b{)}^6=a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6$  from the row 1, 6, 15, 20, 15, 6, 1

Example 1

Expand $(2x+3{)}^3$

Let $a=2x$ and $b=3$

Then,

$$\begin{array}{rl}(2x+3{)}^3& =(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3\\ \\ (2x+3{)}^3& =(2x{)}^3+3(2x{)}^2(3)+3(2x)(3{)}^2+3^3\\ \\ (2x+3{)}^3& =8x^3+3(4x^2)(3)+3(2x)(9)+27\\ \\ (2x+3{)}^3& =8x^3+36x^2+54x+27\end{array}$$

Example 2

Expand $(3m-5{)}^5$

Let $a=3m$ and $b=-5$

Then,

$$\begin{array}{rl}(3m-5{)}^5& =(a+b{)}^5=a^5+5a^{4}b+10a^3{b}^2+10a^2{b}^3+5a{b}^4+b^5\\ \\ (3m-5{)}^5& =(3m{)}^5+5(3m{)}^4(-5)+10(3m{)}^3(-5{)}^2+10(3m{)}^2(-5{)}^3+5(3m)(-5{)}^4+(-5{)}^5\\ \\ (3m-5{)}^5& =243m^5+5(81m^4)(-5)+10(27m^3)(25)+10(9m^2)(-125)+5(3m)(625)-3125\\ \\ (3m-5{)}^5& =243m^5-2025m^4+6750m^3-11250m^2+9375m-3125\\ \end{array}$$