SOLUTION: how many ways can three awards be given to 6 students. each student may receive more than one award? each student may receive no more than one award?

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Question 1200865: how many ways can three awards be given to 6 students. each student may receive more than one award? each student may receive no more than one award?
Answer by ikleyn(52779) About Me  (Show Source):
You can put this solution on YOUR website!
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In how many ways can three awards be given to 6 students if
(a) each student may receive more than one award?
(b) each student may receive no more than one award?
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            From the context,  the students are distinguishable,
            but the awards are indistinguishable. 


(b)  It is more easy to start from the second question.

     It is the same as to ask, in how many different ways 3 students can be selected from 6 students ?

     It is a classic problem on COMBINATIONS, and the ANSWER is

             in  C%5B6%5D%5E3 = %286%2A5%2A4%29%2F%281%2A2%2A3%29 = 5*4 = 20 different ways.



(a)  It is the same as to ask how many solutions this equation

         x%5B1%5D + x%5B2%5D + x%5B3%5D + x%5B4%5D + x%5B5%5D + x%5B6%5D = 3   (1)

     has in integer non-negative numbers.


     It is a classic problem to solve using the "stars and bars" method.


     The answer is: there are  %283%2B6-1%29%21%2F%283%21%2A%286-1%29%21%29 = 8%21%2F%283%21%2A5%21%29 = %288%2A7%2A6%29%2F%281%2A2%2A3%29 = 8*7 = 56
     different solutions for equation (1) in non-negative integer numbers.

     So, there are 56 different ways to award 6 students by 3 awards, if each student may receive more than one award 

     (the awards are considered as undistinguishable; the students are distinguishable).

Solved.

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For the stars and bars method see this Wikipedia article
https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

or read my lesson
    - Stars and bars method for Combinatorics problems
in this site.