Question

in how many ways could 18 people be divided into five groups containing, respectively, 2, 6, 4, 1, and 5 people? the groups can be chosen in ways.

Explanation:

Step1: Use the multinomial coefficient formula

The formula for dividing $n$ objects into $k$ groups of sizes $n_1,n_2,\cdots,n_k$ (where $n = n_1 + n_2+\cdots + n_k$) is $\frac{n!}{n_1!n_2!\cdots n_k!}$. Here, $n = 18$, $n_1=2$, $n_2 = 6$, $n_3=4$, $n_4 = 1$, $n_5=5$.

Step2: Calculate the factorial values

We know that $n!=n\times(n - 1)\times\cdots\times1$. So, $18! = 18\times17\times\cdots\times1$, $2! = 2\times1$, $6! = 6\times5\times4\times3\times2\times1$, $4! = 4\times3\times2\times1$, $1! = 1$, $5! = 5\times4\times3\times2\times1$.

Step3: Compute the result

The number of ways is $\frac{18!}{2!6!4!1!5!}=\frac{18\times17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7\times6!}{2\times1\times6!\times4\times3\times2\times1\times1\times5\times4\times3\times2\times1}$.
After simplification:
\[

$$\begin{align*} &\frac{18\times17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7}{2\times24\times120}\\ =&\frac{18\times17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7}{2\times24\times120}\\ =&122412240 \end{align*}$$

\]

Answer:

$122412240$