Q:

How to deduce the formula for arrangement in groups?

The number of ways in which $n$ different things can be arranged into $r$ different groups is $$n! \cdot \dbinom{n - 1}{r - 1} $$.

This is the quote from my book. However, it didn't offer any explanation. Now, how is it deduced?

A:

The easiest way to look at this is to line up your $n$ elements
\begin{equation}
e_1e_2\cdots e_n
\end{equation}
Now we want to divide this particular sequence into $r$ groups. That means that we need to pick $r-1$ spaces between elements to divide. One example of $5$ elements into $3$ groups is
\begin{equation}
e_1e_2|e_3|e_4e_5
\end{equation}
Note that there are $n-1$ possible spaces to choose from. Therefore, for a particular sequence we get $\binom{n-1}{r-1}$ choices for the placement of the divisors. Now, if we count two groups different if the elements in them are placed in a different order, then we just need to multiply by the total number of line ups, which equals $n!$.

