The are $5$ identical white balls and $2$ identical black balls in the box. In how many ways can you draw $1$ black and $2$ white balls?
[Math] In how many ways can you draw 3 balls
combinatorics
combinatorics
The are $5$ identical white balls and $2$ identical black balls in the box. In how many ways can you draw $1$ black and $2$ white balls?
Best Answer
There are two black balls, and we want to choose one: $\dbinom 21 = 2$.
There are five white balls from which we want to draw two: $\dbinom 52 = \dfrac{5!}{2!\,3!} = \dfrac{5\cdot 4}{2} = 10$..
We use the rule of the product (multiplying) to obtain the total number of ways of choosing one black ball and two white balls:
That gives us $$\binom 21 \cdot \binom 52 = 2\cdot \dfrac{5!}{2!3!} = 2\cdot \dfrac{5\cdot 4}{2} = 20$$