Hello all!

Probably one of the most difficult things for me in a high school math course is the application of permutations and combinations to real life problems. I just always, always, no matter how much I practice, get the wrong.

The review book I am using introduces the following formula for arrangements with repetitions.

It says:

The number of different arrangements of size $\displaystyle m$ chosen out of $\displaystyle k$ groups of similar elements is given by

$\displaystyle

\frac{(k+m-1)!}{m!(k-1)!}

$

I just don't get this formula. For example, consider the following example that the book has

A new train schedule is being made with the following requirements, on three days of the week there must be two departures, on two days of the week one departure, and on two days three departures. How many different schedules is it possible to make?

So if we use the formula above we have k-the number of groups-equals 3, and m equals 7, but the result turns out to be completely weird. Otheriwise, you could solve the problem as $\displaystyle \frac{7!}{2!3!2!}$. But I still don't understand the book's formula.

All help, comments and input is appreciated.