number of ways in which it is possible to make a selection from m+n+p=N things,where p are alike of one kind, m alike of another kind and n alike of another kind,taken r at a time is given by co-efficient of x^r in the expansion of
(1+x+x^2+x^3+......+x^m)(1+x+x^2+.....x^n)(1+x+x^2 +x^3+.......x^p).
How is this established?Explain me the Q.E.D.
A selection ofobjects is determined by the number of objects of each kind. So, the number of such selections is the number of triples
such that
and
,
and
. When you look at the polynomial, this is exactly the coefficient of
. Indeed, to get
from the first factor,
from the second factor and
from the third factor so that
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