Answer
The number of arrangements that can be made using four of the letters of the word COMBINE is 840.
Work Step by Step
We need to choose 4 letters from a group of 7 letters. The order in which the letters are chosen matters because 7 letters of the word "COMBINE" are distinct. Therefore, we need to find the number of permutations of 7 things taken 4 at a time.
Apply the formula,
${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$
Where $ n=7,r=4$
$\begin{align}
& {}_{7}{{P}_{4}}=\frac{7!}{\left( 7-4 \right)!} \\
& =\frac{7!}{3!}
\end{align}$
Solve further:
$\begin{align}
& {}_{7}{{P}_{4}}=\frac{7\times 6\times 5\times 4\times 3!}{3!} \\
& =7\times 6\times 5\times 4 \\
& =840
\end{align}$
