M is number of times 6 occurs on the dice.
Hence Probability that out of n throws m times 6 occurs will be :
Choose m throws out of n throws with ^{n}C_{m} ways where 6 comes and then 6 comes with probability 1/6 in those m places and remaining places 6 does not come with 5/6 probability.
So, P {M = m} will be given by the below expression :
P{M = m} = ^{n}C_{m} . (1/6)^{m} . (5/6)^{(n-m) }
This is a Binomial Distribution with p = 1/6.
Hence, EM = ∑_{m=0 to n }m. ^{n}C_{m} . (1/6)^{m} . (5/6)^{(n-m)}
Basically, EM = sum from m=0 to m=n of ( m . P{M = m})
We already know that this summation will simplify to np, the derivation of Expectation of a Binomial Distribution can be looked up in any standard book.
So, EM = np = n/6.