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'strange' simplification problems - why?


> Product[lmb(M-i)/(mu(i+1)),{i,0,k-1}]

produces


> ((-1)^k*lmb^k*(mu^(-1))^k Pochhammer[-M, k])/k!

there are two things wrong with that for me:

a)  lmb^k*(mu^(-1))^k  is just a longwinded way of saying (lmb/mu)^k -
and I get the long form also after FullSimplify. What am I doing wrong
here? how could I get that to the simpler form? [short of replacing it
by hand]

b) more difficult: the 'real' answer to the original problem is
(lmb/mu)^k Binomial[M, k]   - which is preferable as it avoids the
Gamma function. How can I coach Mathematica to deliver answers in Binomial
coefficients rather than in Gammas?


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