Question

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial :$a^3{b}^4$term of${(2a+b)}^7$.

Short answer

The coefficient of $a^3{b}^4$term in the expansion of ${(2a+b)}^7$ is$280$.

Step-by-step solution

Step 1. Given 

A binomial is given as ${(2a+b)}^7$.

To find the coefficient of the term $a^3{b}^4$in the expansion of the given binomial.

Step 2. Compare binomial (a+b)n and  (2a+b)7.

On Comparing, we get

$$\begin{gathered} a=2a \\ b=b \\ n=7 \end{gathered}$$

Now in the formula ,

$$\begin{gathered} n-r=3 \\ 7-r=3 \\ r=4 \end{gathered}$$

Step 3. Substitute the obtained values in the formula Crnan-rbr.

We get

$$\begin{gathered} {}^{7}C_4{\left(2a\right)}^{7-4}{b}^4=\frac{7{\displaystyle !}{\displaystyle }}{(7-4){\displaystyle !}{\displaystyle }{\displaystyle 4}{\displaystyle !}{\displaystyle }}\left(8a^3\right)b^4 \\ =35\left(8a^3\right)b^4 \\ =280a^3{b}^4 \end{gathered}$$

The coefficient of the term$a^3{b}^4$for the given binomial is$280$.