There are $k$ sets of numbers: $$\lbrace0,1,2,\ldots,m_1\rbrace, \lbrace0,1,2,\ldots,m_2\rbrace, \ldots,\lbrace0,1,2,\ldots,m_k\rbrace$$ Such that $m_1 \lt m_2 \lt \cdots \lt m_k$.

How many combinations of k elements can be made taken 1 element from each set such that each set has all distinct elements (no two elements are equal)?

What is the probability that any two sets will have at least one element common?

I have found that the number of such permutations will be $(m_1+1)m_2(m_3-1)......(m_k-k+2)$, but what will be the number of combinations? Clearly it cannot be just = No. of permutations/$k!$.