1. **Problem Statement:** We want to find the number of ways to arrange 8 objects into 3 places. This is a permutation problem because the order matters. 2. **Formula Used:** The number of permutations of $n$ objects taken $r$ at a time is given by: $$P(n,r) = \frac{n!}{(n-r)!}$$ where $n!$ (n factorial) means the product of all positive integers from $n$ down to 1. 3. **Understanding Factorials:** - $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$ - For example, $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ - Factorials grow very fast as $n$ increases. 4. **Applying the Formula:** - Here, $n=8$ and $r=3$. - So, $P(8,3) = \frac{8!}{(8-3)!} = \frac{8!}{5!}$. 5. **Expanding Factorials:** - $8! = 8 \times 7 \times 6 \times 5!$ - So, $\frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5!}{5!}$ 6. **Canceling Common Factors:** - Cancel $5!$ in numerator and denominator: $$\frac{8 \times 7 \times 6 \times \cancel{5!}}{\cancel{5!}} = 8 \times 7 \times 6$$ 7. **Calculating the Result:** - Multiply the remaining numbers: $$8 \times 7 = 56$$ $$56 \times 6 = 336$$ 8. **Final Answer:** $$P(8,3) = 336$$ This means there are 336 different ways to arrange 8 objects into 3 places when order matters.