Sum of the variance of individual element of multinomial distribution

Question

Suppose I have a multinomial distribution with 10 outcomes and each has prob of 0.1 (i.e. weight of 10%). After each random draw, I can calculate the weight of the sample w1, w2... w10, and I want to calculate the expected deviation from the mean: $E((w_1-0.1)^2 + (w_2-0.1)^2... + (w_10-0.1)^2)$. Somehow I feel like I can reduce it to summing the variance of each individual outcome. But what happen to the covariance term? https://en.wikipedia.org/wiki/Multinomial_distribution I am confused and feel I am missing something.