A volunteer coordinator has 30 identical chocolate chip cook

A volunteer coordinator has 30 identical chocolate chip cookies to distribute to six volunteers. Use a generating function (and computer algebra system) to determine the number of ways she can distribute the cookies so that each volunteer receives at least two cookies and no more than seven cookies.

Solution

Chocolate chip cookies identical = 30

Volunteers = 6

Each volunteer receives atleast 2 and not more than 7

As all chip cookies are identical order does not matter.

First distribute minimum 2 compulsory to all 6 volunteers.

Now remaining are = 30-12 = 18 cookies

Already each one is having 2. Hence each one can receive at most 5 from the remaining

Hence it can be distributed as (5,5,5,3) ... 6C3 (3C1) ways

                         Or                  (5,5,4,4) ...     6C2(3C2) ways

                         Or                   (5,5,5,2,1) ...     6C3(3C2) ways

                         Or (5,5,4,3,1) ...     6C2(3C2) ways

                         Or   (4,4,4,2)