I came across the following problem in a exam. 

We choose a permutation of n elements $[1,n]$ uniformly at random. Now a variable MIN holds the minimum value seen so far at it is defined to $\infty$ initially. Now during our inspection if we see a smaller value than MIN, then MIN is updated to the new value. 

For example, if we consider the permutation, 

$$5\ 9\ 4\ 2\ 6\ 8\ 0\ 3\ 1\ 7$$
the MIN is updated 4 times as $5,4,2,0$. Then the expected no. of times MIN is updated?

I tried to find the no. of permutations, for which MIN is updated $i$ times, so that I can find the value by $\sum\_{i=1}^{n}iN(i)$, where $N(i)$, is the no. of permutations for which MIN is updated $i$ times.

But for $i\geq2$, $N(i)$ is getting very complicated and unable to find the total sum.