I'm having trouble figuring out exactly how one would get the answer for a question such as:
"how many ways can 9 concert tickets be divided between 4 concertgoers, such that one person receives 3 tickets, and the rest receive 2 each?"
The answer provided is:
$\displaystyle\frac{9!}{3!2!2!2!} = 7560$
So, I'm having trouble finding exactly what the formula for this is - When I google 'Ordered Partitions' I can't see a formula that I could use to form an expression based on the english language description of the problem.
So, my thinking would be the answer could be derived from finding $n$ factorial, where $n$ is the total number of the items, divided by the factorial of some other constants(factorial), where all the constants add up to $n$ (disregarding the factorial) e.g. n = 9, and our constants are 3 + 2 + 2 + 2 = 9, thus the formula might be e.g. $n!/a!*b!*c!$
So if we had a question for example: "There are 21 marbles in a bag, and five people choosing marbles. How many ways can the marbles be divided between these people so that one person receives 7, one receives 5, and the rest 3 each?
Would the correct way to go about it be to find the constants... 5 people... one receives 7, one 5, and the rest 3 each. so 7 + 5 + 3 +3 + 3 = 21. This equals to the total number $n$, so we do the factorial of$n$ over 7! + 5! + 3! + 3! + 3!
...
$\displaystyle\frac{21!}{7! \times 5! \times 3! \times 3! \times 3!} = 391091500800$
If anyone could provide a formula for this concept, let me know if my understanding is correct or if not, why - and if possible provide some documentation for this concept, I would greatly appreciate it.
Thanks!
You are correct, both in your setup at the end and in your explanation of the derivation. This is also called a multinomial coefficient, the generalization of a binomial coefficient, and the idea is much the same. With a binomial coefficient, we want to know how many ways there are to distribute $n$ things, with $k$ going to the first person and $n-k$ going to the second.
In a binomial coefficient $\binom{n}{k}$, we write only one number at the bottom, but the other number ($n-k$) is implied. With a multinomial coefficient, we write everything, e.g.,
$$\binom{21}{7,5,3,3,3}$$