# How do I use Pascal's triangle to expand (3a + b)^4?

##### 1 Answer
Jul 3, 2015

Combine a row of Pascal's triangle with a list of powers of $3$ to find:

${\left(3 a + b\right)}^{4} = 81 {a}^{4} + 108 {a}^{3} b + 54 {a}^{2} {b}^{2} + 12 a {b}^{3} + {b}^{4}$

#### Explanation:

Write out the 5th row of Pascal's triangle as a sequence:

$1 , 4 , 6 , 4 , 1$

Write out powers of $3$ from ${3}^{4}$ down to ${3}^{0}$ as a sequence:

$81 , 27 , 9 , 3 , 1$

Multiply the two sequences together to get the sequence:

$81 , 108 , 54 , 12 , 1$

These are the coefficients of the expansion:

${\left(3 a + b\right)}^{4} = 81 {a}^{4} + 108 {a}^{3} b + 54 {a}^{2} {b}^{2} + 12 a {b}^{3} + {b}^{4}$