Question

In how many ways can 3 blue, 4 white and 2 red balls be distributed into 4 distinct boxes?

• A. 49000
• B. 36750
• C. 126
• D. 7000

Hint:

In problems of distributing identical objects into distinct boxes, use the stars and bars formula for each group and then multiply the results.

Answer 1

The Correct Option is D

This is a problem of distributing distinct objects (balls) into distinct boxes, where the objects of the same color are identical. The formula for the number of ways to distribute \(n\) identical objects into \(r\) distinct boxes is given by: \[ \frac{(n+r-1)!}{n!(r-1)!} \] We calculate separately for each color of balls: - For blue balls: There are \(3\) blue balls and \(4\) boxes, so the number of ways is: \[ \frac{(3+4-1)!}{3!(4-1)!} = \frac{6!}{3!1!} = 20 \] - For white balls: There are \(4\) white balls and \(4\) boxes, so the number of ways is: \[ \frac{(4+4-1)!}{4!(4-1)!} = \frac{7!}{4!3!} = 35 \] - For red balls: There are \(2\) red balls and \(4\) boxes, so the number of ways is: \[ \frac{(2+4-1)!}{2!(4-1)!} = \frac{5!}{2!3!} = 10 \] The total number of ways is the product of these individual results: \[ 20 \times 35 \times 10 = 7000 \]