Question

in how many ways could 17 people be divided into five groups containing, respectively, 7, 4, 2, 3, and 1 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 $n=17$, $n_1 = 7$, $n_2=4$, $n_3 = 2$, $n_4=3$, $n_5 = 1$.

Step2: Calculate the factorial values

$n!=17!$, $n_1!=7!$, $n_2!=4!$, $n_3!=2!$, $n_4!=3!$, $n_5!=1!$.
$17! = 17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7!$, $4! = 4\times3\times2\times 1$, $2! = 2\times1$, $3! = 3\times2\times1$, $1! = 1$.

Step3: Substitute and simplify

$\frac{17!}{7!4!2!3!1!}=\frac{17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7!}{7!\times(4\times3\times2\times1)\times(2\times1)\times(3\times2\times1)\times1}$
$=\frac{17\times16\times15\times14\times13\times12\times11\times10\times9\times8}{4\times3\times2\times1\times2\times1\times3\times2\times1}$
$=12376\times13\times11\times9\times8$
$=136136640$.

Answer:

$136136640$