Question 1204253
(a)  Find the number of ways that Magnus can give out 12 identical stickers 
     to 12 of his friends. (Not everyone has to get a sticker.)
(b)  Find the number of ways that Magnus can give out 12 identical stickers 
     to 12 of his friends, if every friend gets at least one sticker.
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<pre>
                 <U>Part (a)</U>



In this problem, stickers are not distinguishable, but the friends are.

This problem is equivalent to the question: how many solutions does this equation have

    {{{x[1]}}} + {{{x[2]}}} + {{{x[3]}}} + . . . + {{{x[k]}}} = n

in integer non-negative numbers at n= 12, k= 12 ?


The answer is: the number of such solutions is {{{C[n+k-1]^n}}} = {{{C[12+12-1]^12}}} = {{{C[23]^12}}} = 1352078.


This formula is deduced using so called "stars and bars method".


On "stars and bars method", see this Wikipedia article 

https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

or read from my lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Stars-and-bars-method-for-Combinatorics-problems-2.lesson>Stars and bars method for Combinatorics problems</A> 

in this site.



                 <U>Part (b)</U>



This problem means that each friend gets exactly one sticker.


And since the stickers are undistinguishable, it means that there is only one scenario 
to distribute stickers in a way, as it is assigned by the problem.
</pre>

Solved.



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Does the answer to (b) seem to be unexpected ?   &nbsp;&nbsp;&nbsp;&nbsp;Did you expect to get the answer 12! ? 



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Remember that 12! relates to permutations of 12 different items, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;while in this problem all the stickers are identical.



This problem teaches you to read the problem attentively.