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Chapter 7: Problem 362

How many ways can \(\mathrm{r}\) different balls be placed in n different boxes? Consider the balls and boxes distinguishable.

Short Answer

Expert verified
The total number of ways to place \(r\) different balls in \(n\) different boxes, where balls and boxes are distinguishable, is \(n^r\).

Step by step solution

01

Apply the Multiplication Rule

Since we have r balls and n boxes, there are n choices for the first ball, n choices for the second ball, and so on, until n choices for the r-th ball. To find the total number of possible placements, we simply need to multiply the number of choices for all r balls: \( Total \: Placements = n * n * ... * n \) (r times)
02

Simplify the Expression

As there are r factors of n, we can write the total number of placements more compactly using exponentiation: \( Total \: Placements = n^r \)
03

Interpret the Result

The total number of ways to place r different balls in n different boxes, where balls and boxes are distinguishable, is \(n^r\).

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