1. Problem: Find the number of permutations of 3 objects taken all at a time. Formula: The number of permutations of $n$ objects taken $r$ at a time is given by $$P(n,r) = \frac{n!}{(n-r)!}$$ Step 1: Here, $n=3$ and $r=3$. Step 2: Calculate $$P(3,3) = \frac{3!}{(3-3)!} = \frac{3!}{0!} = \frac{6}{1} = 6$$ Answer: There are 6 permutations. 2. Problem: Find the number of permutations of 5 objects taken 2 at a time. Step 1: $n=5$, $r=2$. Step 2: Calculate $$P(5,2) = \frac{5!}{(5-2)!} = \frac{120}{6} = 20$$ Answer: 20 permutations. 3. Problem: How many ways can 4 books be arranged on a shelf? Step 1: $n=4$, $r=4$. Step 2: Calculate $$P(4,4) = 4! = 24$$ Answer: 24 ways. 4. Problem: Find permutations of 6 objects taken 3 at a time. Step 1: $n=6$, $r=3$. Step 2: Calculate $$P(6,3) = \frac{6!}{3!} = \frac{720}{6} = 120$$ Answer: 120 permutations. 5. Problem: Number of ways to arrange 7 people in a line. Step 1: $n=7$, $r=7$. Step 2: Calculate $$P(7,7) = 7! = 5040$$ Answer: 5040 ways. 6. Problem: Permutations of 8 objects taken 4 at a time. Step 1: $n=8$, $r=4$. Step 2: Calculate $$P(8,4) = \frac{8!}{4!} = \frac{40320}{24} = 1680$$ Answer: 1680 permutations. 7. Problem: Find permutations of 10 objects taken 1 at a time. Step 1: $n=10$, $r=1$. Step 2: Calculate $$P(10,1) = \frac{10!}{9!} = 10$$ Answer: 10 permutations. 8. Problem: Number of ways to arrange 9 objects taken all at a time. Step 1: $n=9$, $r=9$. Step 2: Calculate $$P(9,9) = 9! = 362880$$ Answer: 362880 ways. 9. Problem: Permutations of 5 objects taken 3 at a time. Step 1: $n=5$, $r=3$. Step 2: Calculate $$P(5,3) = \frac{5!}{2!} = \frac{120}{2} = 60$$ Answer: 60 permutations. 10. Problem: Number of ways to arrange 12 objects taken 2 at a time. Step 1: $n=12$, $r=2$. Step 2: Calculate $$P(12,2) = \frac{12!}{10!} = \frac{479001600}{3628800} = 132$$ Answer: 132 permutations.