Combinatorics: Confusion in identical objects in distinct groups I'm confused between the following $2$ formulae:
1) Number of ways to put $n$ identical objects into $r$ distinct boxes, such that the ordering is NOT important is:
  $$\binom{n+r-1}{n}$$
2) Number of ways to distribute $n$ candies to $r$ children (distinct, obviously) is:
$$r^n$$
Aren't the two statements equivalent ? So, what makes their formulae different ?
 A: Assume we have $n=4$ candies, a red, a blue, a green and a yellow one. We have $r=2$ children, Alice and Bob.
In (1) (assuming you fix the $r$ to be an $n$ in the binomial coefficient) you count how to distribute the candies between Alice and Bob, when the color doesn't matter. So you won't count Alice getting blue and red, Bob getting green and yellow separately from Alice getting blue and yellow, Bob getting green and red. You just count the tuples $(a,b)$ where $a$ is the number of candies Alice gets, $b$ is the number of candies Bob gets.
In (2) you do count those combinations as two. Here you count quadruples $(x,y,z,w)$ where $x,y,z,w\in\{\text{Alice}, \text{Bob}\}$ and $x$ denotes who got the red candy, $y$ who got the blue candy, etc.
A: Edit: G-man and H_T, I'm afraid your answers are not right. This is the comprehensive list of all cases.
Case 1: Distinct candies and distinct children.
Ordering important
n(n+1)(n+2)....(n+r-1)
Ordering not important
$$r^n$$
Case 2: Identical candies and distinct children.
Ordering important:
Case does not arise, since candies are identical.
Ordering not important:
$$\binom{n+r-1}{n}$$
So the second statement about distributing candies to children, is a case of distinct objects in distinct boxes, with ordering not important. That's why the formulae are different.
