how many ways are there to share 12 white ball and 3 red ball between 5 person in case that...

Question

How many ways are there to share 12 white balls and 3 red balls between 5 person in case that :

a) every person should have at least one ball and non of them should receive more than one red ball.

b) every person should have at least one ball and all the red balls should be shared with only 2 persons.

please help me do these questions, i think they are related to permutation with the repetition but i do not know how to solve them.

Answer 1

For a):

• Ways of choosing $3$ of the $5$ people and giving them one red ball each: $\color{blue}{\binom 53}$
• Ways of giving the remaining $2$ people one white ball: $\color{blue}{1}$
• Ways of distibuting the remaining $10$ white balls among the $5$ people: $\color{blue}{\binom{10+4}{4}}$

All together:

$$\color{blue}{\binom 53\cdot \binom{10+4}{4}}$$

For b):

• Ways of choosing $2$ of the $5$ people and distributing the $3$ red balls, so that each has at least one: $\color{blue}{\binom 52\cdot 2}$
• Ways of giving the remaining $3$ people one white ball: $\color{blue}{1}$
• Ways of distibuting the remaining $9$ white balls among the $5$ people: $\color{blue}{\binom{9+4}{4}}$

$$\color{blue}{\binom 52\cdot 2 \cdot \binom{9+4}{4}}$$ $\color{blue}{}$