How many compilers ie front ends and back ends are required for m languages and n machines by making use of an intermediate language?

If the intermediate code generated is the same for all the machines. Hence the

number of Front end conversions is one for each language. Similarly the number of

back end conversions is equal to the number of machines.

Intermediate representation has a variety of forms. There are also many algorithms for

generating intermediate codes for typical programming language constructs.

To generalize, for M languages on N machines we would require M Front end

conversions. The number of back end conversions would then be equal to N*M.