Clarification of surjections

[mahjk17]

I am studying surjections in my combinatorics class and in the book it states, " A teacher has 15 distinct candy bars and wants to distribute the distinct candy bars to 4 of her students so that each of the student gets at least one candy bar, how many ways can she make that distribution"? The answer to this question is S(15, 4) = 1016542800. Shouldn't the answer be a lot less than that? Isn't this question about enumerating integer solutions to an equation? For some reason a surjection will give you a higher amount to the answer? Can someone please clarify this to me??

 

[HallsofIvy]

The critical point is the word "distinct". There are 15 distinct candy bars so that giving the "Hershey bar" to student A and the "Mars bar" to student B is different from giving the "Mars bar" to student A and "Hershey bar" to student B. It is counting all of the ways of distributing those 15 different bars to different students that gives that large answer.

 

[mahjk17]

Thank you very much !!! : )