Why is this way of calculating variance incorrect?

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Each morning John eats some eggs. On any given morning, the number of eggs he eats is equally likely to be 1, 2, 3, 4, or 5 independent of what he has done in the past. Let X be the number of eggs that John eats in 10 days. Find the mean and variance of X.


I didn't have trouble with the mean:

$E(X) = [\frac{1 + 2 + 3 + 4 + 5}{5}] * 10 = 3 * 10 = 30$ (multiplying by 10 because of 10 days)

Now for the variance...

$var(X) = E(X^2) - (E(X))^2$

$E(X^2) = [\frac{1^2 + 2^2 + 3^2 + 4^2 + 5^2}{5}] * 10^2 = 11 * 100 = 1100$ (multiplying by 10 because of 10 days, right?)

$var(X) = 1100 - 30^2 = 200$

:)

edit: Solved