In how many ways can b blue balls and r red balls be distributed in n distinct…

Assumption: balls of the same color are identical; boxes are distinct and may hold any number of balls.

Step 1: Count distributions of blue balls.

The number of ways to place b identical blue balls into n distinct boxes (allowing zero in a box) is

• C(n+b-1, b) = (n+b-1)!/(b!(n-1)!)

Step 2: Count distributions of red balls.

Similarly, the number of ways to place r identical red balls into n distinct boxes is

• C(n+r-1, r) = (n+r-1)!/(r!(n-1)!)

Step 3: Combine the counts.

Placements of blue and red balls are independent, so multiply the two counts to get the total number of distributions:

• Total = C(n+b-1, b) × C(n+r-1, r) = (n+b-1)!(n+r-1)!/[(n-1)! b! (n-1)! r!]

Final answer: (n+b-1)!(n+r-1)!/[(n-1)! b! (n-1)! r!].