Distinction between distinct and alike objects. What difference does it make when I have say $15$ toys to be distributed among $3$ children subject to the condition that any child can take any number of toys:-
$a)$ When toys are distinct.
$b)$ When toys are alike.
CONFUSION:-
$a)$ Since any child can get any number of toys the first toy can be distributed in $3$ ways,the second toy can be distributed in $3$ ways and so on till $15$th toy i.e. a total of $3^{15}$ ways.
$b)$I know this has to be done by Multinomial theorem ultimately by solving the equation $x+y+z=15$ for non negative integers $x,y,z$ .But why doesn't the same logic apply here also?What's the basic difference we have to account for when toys are alike instead of distinct?
OR , 
how do we justify that the answer to $b)$ is not $3^{15}$.

Look at another approach if you get confused with it. Draw $15$ alike toys(suppose dots) on a paper . Now you just want to distribute them including $0$ so now we create gaps with a stroke  . Now 15 toys create 17 gaps including the terminals as $0$ can also be the one of the solutions. Two strokes will create three sections so you just have to select any two places from $17$ and combination operator gives the answer so its ${17 \choose 2}$ hope you get this approach. See the toys are same . So any order dosent matter . Like child $1$ can get 1st,2nd toy or 1st,20th toy its the same thing as toys are the same.
