A fair die is tossed successively. Let $X$ denote the number of tosses until each of the six possible outcomes occurs at least once. Find the probability mass function of $X$. I'm also given this $hint$: For $1\leq i \le6$ let $E_i$ be the event that the outcome $i$ does not occur during the first $n$ tosses of the die. First calculate $P(X>n)$ by writing the event $X>n$ in terms of $E_1, E_2,...E_6$.
I know that $P(X>n)=1-P(X
I looked and the answer is $(\frac56)^{n-1}-5(\frac46)^{n-1}+10(\frac36)^{n-1}-10(\frac26)^{n-1}+5(\frac16)^{n-1}\quad for \quad n\ge6$ I tried to derive how this was found but I found the alternating signs to be tricky and I'm also confused with why the coefficients are what they are.