What is the probability of the product rolled with 4 different dice being 24?

Question

Four ways to roll 24:

• 1 1 4 6
• 1 2 2 6
• 1 2 3 4
• 2 2 3 4 2 2 2 3

Since the dice are different, there are 4! ways to arrange each way, so $\frac{4*4!}{6^4} = \frac{2}{27}$?

Answer 1

Hint:

First of all, the only ways to roll a 24 product are: $1, 1, 4, 6$ and $1, 2, 2, 6$ and $2, 2, 2, 3$ and $1, 2, 3, 4$.

Second thing - there are not 4! ways to arrange for example $1,1,4,6$ or $1,2,2,6$. There are only $12$ ways to do so (you have to pick a place for two numbers and then the remaining spots get filled in).

Similarly there are only $4$ ways of arranging $2, 2, 2, 3$.

Answer 2

The "ways to arrange" must take into account the undistinguishable outcomes

To count them, say you have $n$ objects, divided into $k$ disjoint classes of undistinguishable objects, and say that class $C_i$ ($0≤i<k$) has size $m_i$, where $\sum_{i<k} m_i = n$, the number of different arrangements is $$ \frac{n!}{\prod m_i!} $$

When all objects are distinct, this is the same as $n!$, but in the case that you have two equal objects, the result is $\frac{n!}{2}$.

Also you forget $(2,2,2,3)$, which has $\frac{4!}{3!} = 4$ ways to arrange.