<s>[INST] Probability of a random Permutation Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last two.
My try: I mean there will be 5! different combination possible, do I have to look at each of it?
 [/INST] Hint: the permutations you want are just :
$ (...,5,4)$, $(...,5,3)$, $(...,4,5)$, $(...,3,5)$, $(...,2,5)$ and $(...,2,5)$ and you don't care about the order of the other elements where I put the dots.
So the probability is:
$$\frac{1}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{1}{4} + \frac{1}{5}\cdot\frac{1}{4}.$$
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