Answer
False. The correct statement is: to find the fifth term in the expansion of ${{\left( 2x+3y \right)}^{7}}$, we use the formula for finding a particular term with $ r=\underline{4}$, $ a=2x $, $ b=3y $ and $ n=7$.
Work Step by Step
We know that by using the Binomial Formula for the ${{\left( r+1 \right)}^{th}}$ term in the expansion of ${{\left( a+b \right)}^{n}}$, we get:
Term ${{\left( r+1 \right)}^{th}}={}_{n}{{C}_{r}}{{a}^{n-r}}{{b}^{r}}$
To find the fifth term, put $ r=4$
Therefore, by putting $ r=4,a=2x,b=3y $,
The fifth term is,
$\begin{align}
& {{\left( 4+1 \right)}^{th}}={{5}^{th}} \\
& ={}_{7}{{C}_{4}}{{\left( 2x \right)}^{7-4}}{{\left( 3y \right)}^{4}} \\
& ={}_{7}{{C}_{4}}{{\left( 2x \right)}^{3}}{{\left( 3y \right)}^{4}} \\
& ={}_{7}{{C}_{4}}{{\left( 2x \right)}^{7-4}}{{\left( 3y \right)}^{4}} \\
& ={}_{7}{{C}_{4}}8{{x}^{3}}\times 81{{y}^{4}} \\
& ={}_{7}{{C}_{4}}648{{x}^{3}}{{y}^{4}}
\end{align}$
But in the statement for finding the fifth term, the value of $ r=5$, and that is the incorrect value.
Thus, the provided statement is false.
