Question

6. Suppose a small elementary school class has 12 students, and the teacher has 20 pieces of chocolate to give out. How many ways can the teacher distribute all the chocolate to the class if (15 points) (No need to calculate / simplify the result, just set up) a) there are no restrictions? b) each child must get at least one piece of chocolate? c) a certain child is to receive exactly 2 pieces of chocolate?

Answer 1

a) Assuming that each chocolate piece is similar, therefore the question of distributing 20 identical chocolates to 12 students is obtained as a multinomial formula of distributing 20 items into 12 distinct groups such that each can get 0 or more item. Therefore using the multinomial formula, we get here the number of ways as:

$= \binom{n+ k-1}{k-1} = \binom{20+12-1}{12-1} = \binom{31}{11}$

Therefore there are 31c11 ways to distribute the chocolates here.

b) As each student has to get at least 1 chocolate, therefore total number of ways here is computed as:

$= \binom{n -1}{k-1} = \binom{20 -1}{12-1} = \binom{19}{11}$

Therefore there are 19c11 ways to distribute the chocolates here.

c) As a certain child is to receive exactly 2 pieces of chocolate, therefore we are to compute the number of ways to distribute 18 chocolates here to 11 students as:

$= \binom{n+ k-1}{k-1} = \binom{18+11-1}{11-1} = \binom{28}{10}$

Therefore there are 28c10 ways to distribution the chocolates here.