We have $5$ fair dice. What is the probability that they don't have all the same number?
Attempt
We must have at least one die that is different of the others. My argument is:
Let distinguish the dice: we give to each dice a letter $a,b,c,d,e$. So, we need to have either $a$ different of every other die, or $b$ different of every other die or $c$ different of every other die or $d$ different of other dice or $e$ different of other dice. So, for $a$ different of every other dice, we hace $6$ possibilities for $b$, $6$ for $c$, $6$ possibilities for $d$, $6$ possibilities for $e$ and $5$ possibilities for $a$. That make $6^45$ possibilities. For $b$ different of the other, we have the same: $6^4\cdot5$ possibilities as well, for $c$ the same... and so at the end, we get a probability of $\frac{6^4\cdot5}{6^5}\cdot5>1$, and thus it's wrong. The final result is $\frac{6^4}{5}\cdot6^5$ but I don't understand why my argument is not correct.
The answer is $\boxed{7770/7776}$
There are only six ways in which the five dice can each have the same number: they can all equal $1$, they can all equal $2$, they can all equal $3$, etc.
But, we want the probability that they don't all have the same number.
There are $6^{5}$ total ways to choose the outcome of the $5$ dice, since there are $6$ outcomes for the first dice, $6$ for the second dice, and so on.
Thus, the number of desirable outcomes equals $6^{5} - 6 = 7770$.
The probability of obtaining a desirable outcome is just the ratio of the number of desirable outcomes to total outcomes. This gives us $7770/6^{5} = 7770/7776,$ which is the answer.