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 combination formula

The number of ways to choose $n_1$ elements from $N$ elements, then $n_2$ from the remaining $N - n_1$ elements and so on is given by the multinomial coefficient formula $\frac{N!}{n_1!n_2!\cdots n_k!}$, where $N=n_1 + n_2+\cdots + n_k$. Here $N = 17$, $n_1=7$, $n_2 = 4$, $n_3=2$, $n_4=3$, $n_5 = 1$.
The formula for the number of ways to partition the people is $\frac{17!}{7!4!2!3!1!}$.

Step2: Calculate factorial values

$17!=17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7!$, so $\frac{17!}{7!4!2!3!1!}=\frac{17\times16\times15\times14\times13\times12\times11\times10\times9\times8}{4!2!3!1!}$.
$4!=4\times3\times2\times1 = 24$, $2!=2\times1=2$, $3!=3\times2\times1 = 6$, $1!=1$.
$\frac{17\times16\times15\times14\times13\times12\times11\times10\times9\times8}{24\times2\times6\times1}$.
$17\times16\times15\times14\times13\times12\times11\times10\times9\times8=1028160\times13\times12\times11\times10\times9\times8=13366080\times12\times11\times10\times9\times8 = 160392960\times11\times10\times9\times8=1764322560\times10\times9\times8=17643225600\times9\times8 = 158789030400\times8=1270312243200$.
$24\times2\times6\times1 = 288$.
$\frac{1270312243200}{288}=4409417510$.

Answer:

$4409417510$