Question

A woman has 14 identical pens to distribute among a group of 10 distinct students. How many ways are there to distribute the 14 pens such that each student gets at least one pencil?

(a) ^15C10

(b) ^10C5 * 11

(c) ^15C8 * 4!

(d) ^13C9

This question was posed to me in an interview for internship.

The query is from Counting topic in chapter Counting of Discrete Mathematics

Answer 1

Right answer is (d) ^13C9

To explain: For this type of problem, n>=r must be true and so according to stars and bars model, the number of possible arrangements of stars and bars is n-1Cr-1 or equivalently, there are ^n-1Cr-1  distributions of n identical objects into r distinct non-empty bins. In this example, there are n = 14 identical objects to be distributed among r=10 distinct bins. Using the above formula, the number of possible distributions is ^13C9.