There are 5 mangoes,10 apples and 15 oranges.Find number of ways to distribute 15 fruits to two persons.
But before that you need to select $\displaystyle 15$ fruits.And then when you distribute them all fruits are not identical i.e there are $\displaystyle 3$ different varieties of fruits.Your formula works only when all objects are alike and of same genre.
This would be number of ways to select $\displaystyle 15$ fruits from given $\displaystyle 30$ fruits and comes out to be $\displaystyle 66$.
Now,what is to be done.
Unless we take into account each selection separately then only can the answer be determined.
Would it be
[30!/(5!10!15!)]*C(30,15)*C(16,15)
Like... the number of permutations of the 30 fruits mulitplied by the number of combinations of size 15 from 30 possible fruits mulitiplied by the number of different ways you can distribute 15 fruits to 2 people...
Way off?
let us pick a collection of 15 fruits:
2 mangoes, 7 apples and 6 oranges.
now, these can be divided among two people in 3*8*7=168 ways
(these include the cases in which one of them don't get anything).
Let us say that this case corresponds to $\displaystyle (3x^2)(8x^7)(7x^6)$
Then all the solution will be given by $\displaystyle \sum [(n+1)x^n][(m+1)x^m][(16-n-m)x^{15-n-m}]$...subject to constraints:
$\displaystyle 0 \leq n \leq 5$
$\displaystyle 0 \leq m \leq 10$
If the reasoning till now is true, then the solution of given problem is the coefficient of $\displaystyle x^{15}$ in
$\displaystyle (1+2x+...+5x^4+6x^5)(1+2x+...+10x^9+11x^{10})(1+2x +...+15x^{14}+16x^{15})$