How to show that there are $\binom{M+K}{M}$ different number of type bins?
The bins contain at most $M$ elements rather than exactly $M$ elements. This is why your formula doesn't work. In order to use your formula, we need to allow for one more type of element, a dummy type, and then we get $\binom{M+(K+1)-1}{M} = \binom{M+K}{M}$.

As an example, suppose that there is one type of element (so $K=1$). In that case, a bin can contain $0,\ldots,M$ elements, in total $M+1 = \binom{M+K}{M}$ choices.