Problem: A fair die is rolled 10 times. Find the expectation of the number of faces which fail to appear in the ten rolls.
I understand I can do this with indicators: X = I1 + I2 + I3 + I4 + I5 + I6, with Ii being 1 when face i fails to appear and 0 other wise. It's easy to see $$P(I_{j}) = (\frac{5}{6})^{10}$$, so $$E(X)=6 \times (\frac{5}{6})^{10}=0.9690$$
However when I tried to calculate it directly, that is, let X be the numer of faces which don't show up. Since at least one face shows up, X is in {0, 1, 2, 3, 4, 5}.
X=5 means 5 faces don't show up, which is equivalent to all ten rolls show a same number, you can choose 1 number from 6 , so $$P(X=5) = (\frac{1}{6})^{10}\times\left(\begin{array}{c}6\\ 1\end{array}\right)$$
likely, $$P(X=4) = (\frac{2}{6})^{10}\times\left(\begin{array}{c}6\\ 2\end{array}\right) - P(X=5) = (\frac{2}{6})^{10}\times\left(\begin{array}{c}6\\ 2\end{array}\right) - (\frac{1}{6})^{10}\times\left(\begin{array}{c}6\\ 1\end{array}\right)$$
$$P(X=3) = (\frac{3}{6})^{10}\times\left(\begin{array}{c}6\\ 3\end{array}\right) - [P(X=5)+P(X=4)] = (\frac{3}{6})^{10}\times\left(\begin{array}{c}6\\ 3\end{array}\right) - (\frac{2}{6})^{10}\times\left(\begin{array}{c}6\\ 2\end{array}\right)$$
$$P(X=2) = (\frac{4}{6})^{10}\times\left(\begin{array}{c}6\\ 4\end{array}\right) - (\frac{3}{6})^{10}\times\left(\begin{array}{c}6\\ 3\end{array}\right)$$
$$P(X=1) = (\frac{5}{6})^{10}\times\left(\begin{array}{c}6\\ 5\end{array}\right) - (\frac{4}{6})^{10}\times\left(\begin{array}{c}6\\ 4\end{array}\right)$$
$$P(X=0) = (\frac{6}{6})^{10}\times\left(\begin{array}{c}6\\ 6\end{array}\right) - (\frac{6}{6})^{10}\times\left(\begin{array}{c}6\\ 6\end{array}\right)$$
So $$E(X) = 5P(X=5) + 4P(X=4) + 3P(X=3) + 2P[X=2] + 1P[X=1] + 0P[X=0] =$$ $$(\frac{1}{6})^{10}\times\left(\begin{array}{c}6\\ 1\end{array}\right) + (\frac{2}{6})^{10}\times\left(\begin{array}{c}6\\ 2\end{array}\right) + (\frac{3}{6})^{10}\times\left(\begin{array}{c}6\\ 3\end{array}\right) + (\frac{4}{6})^{10}\times\left(\begin{array}{c}6\\ 4\end{array}\right) + (\frac{5}{6})^{10}\times\left(\begin{array}{c}6\\ 5\end{array}\right) = 1.2489$$
These two ways don't lead to a same answer. Which one is wrong and why?