<s>[INST] Probability of rolling 5 dice? Say you picked $\{4 5 2 6 5\}$ to show up in the dice rolls. You win if you have 3, 4, or 5 numbers in the correct sequence.


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*3 numbers in correct sequence would be; $\{4 5 2\}$ OR $\{5 2 6\}$ OR $\{2 6 5\}$

*4 numbers in the correct sequence would be; $\{4 5 2 6\}$ OR $\{5 2 6 5\}$

*5 numbers in the correct sequence would be; $\{4 5 2 6 5\}$


What would be the probability of winning? How would I calculate it?
 [/INST] 3 numbers correct: $(\frac{5}{6})^2\cdot\frac{1}{6^3}\cdot\binom{5}{2}$
4 numbers correct: $\frac{5}{6}\cdot\frac{1}{6^4}\cdot\binom{5}{1}$
5 numbers correct: $\frac{1}{6^5}$
total: $\frac{1}{6^5}\bigg(1+5\cdot\binom{5}{1}+5^2\cdot\binom{5}{2}\bigg)=\frac{276}{7776}=\frac{23}{648}$
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