No of ways of selecting at least one object of which p alike of one kind
q alike of one kind and
r alike of one kind is given by
\[(p+1)(q+1)(r+1)-1\]
No of ways of selecting at least one object of which p alike of one kind
q alike of one kind
and rest are of different kind is given by
\[(p+1)(q+1)(2^{n-p-q})\]
Example
In how many ways at least one fruit can be selected from 4 apples, 5 oranges and 3 bananas?
solution
formula for at least one object to be selected \[(p+1)(q+1)(r+1)-1\]
here p=5
q=4
r=3
\[(5+1)(4+1)(3+1)-1\]=119
