10 people are randomly pressing 10 distinct buttons. A button can be pressed multiple times. One person can only choose to press 1 button. What is the expected number of distinct buttons pressed by the 10 people?
Let $X_i$ be the indicator variable for whether the i-th button is pressed. The issue I am having is it appears the indicator variables are dependent? If they were all independent, then the probability of the i-th variable being pressed is $1 - 0.9^{10}$, and so the the solution is simply $10 * E[X_i] = 10 * (1-0.9^{10})$.
How do you solve this problem when the variables are dependent?
The random variable you care about is $X_1 + \cdots + X_{10}$. The equality $$E[X_1 + \cdots + X_{10}] = E[X_1] + \cdots + E[X_{10}]$$ holds even when the $X_i$ are dependent.