Question

In how many ways can we distribute 10 identical looking pencils to 4 studentsso that each student gets at least one pencil?

Answer 1

Answer:

I will give 2 to each and take 2 for me

Answer 2

First let's assume that the pencils and the students are indistinguishable and list the possible distributions.  Then let's account for the fact that the students are distinguishable.

Assuming for the moment that pencils and students are indistinguishable the 9 possible ways to distribute the pencils so that every student gets at least one pencil are:

7-1-1-1

6-2-1-1

5-3-1-1

5-2-2-1

4-4-1-1

4-3-2-1

4-2-2-2

3-3-3-1

3-3-2-2

But we know or assume the students are distinguishable. Thus, for example, in the first distribution listed above, we need to account for the fact that ANY of the 4 students could get the 7 pencils. Similarly, for each of the distributions, we must account for the assignment of the given numbers of pencils to particular students. The number of ways to assign the given numbers of pencils to the students is given by the common permutation formula with repeated objects. The numbers are given in parentheses following the distribution.

7-1-1-1 (4 = 4!/1!3!)

6-2-1-1 (12 = 4! / 1!1!2!)

5-3-1-1 (12 = 4! / 1!1!2!)

5-2-2-1 (12 = 4! / 1!2!1!)

4-4-1-1 (6 = 4! / 2!2!)

4-3-2-1 (24 = 4! / 1!1!1!1!)

4-2-2-2 (4 = 4! / 1!3!)

3-3-3-1 (4 = 4! / 2!1!)

3-3-2-2 (6 = 4! / 2!2!)

Thus, the total number of ways of distributing 10 identical pencils to 4 different students so that every student gets at least one pencil is 84, i.e., the sum of the number of permutations for each distribution. Extending the calculation to other pencil distributions (including those in which one or more students get no pencils), it is seen that the total number of ways of distributing 10 pencils to 4 students without the condition is 286.