1. Problem 17: Arrange 6 distinct people into 2 distinct rows with all people in one of the rows. Formula: Number of ways to assign each person to one of 2 rows = $2^6$. Since rows are distinct and all 6 must be placed, but order within rows matters (assuming arrangement in rows means order matters), we first choose who goes to each row, then arrange them. Step 1: Choose $k$ people for row 1, the rest go to row 2. $k$ can be from 0 to 6. Step 2: Arrange $k$ people in row 1: $k!$ ways. Step 3: Arrange $6-k$ people in row 2: $(6-k)!$ ways. Total ways = $\sum_{k=0}^6 \binom{6}{k} k! (6-k)! = \sum_{k=0}^6 \frac{6!}{k!(6-k)!} k! (6-k)! = \sum_{k=0}^6 6! = 7 \times 720 = 5040$. So total ways = $5040$. 2. Problem 18: Distribute 4 different balls into 2 distinct boxes with no box empty. Total ways without restriction: Each ball has 2 choices, so $2^4=16$. Subtract cases where a box is empty: 2 cases (all balls in box 1 or all in box 2). Total valid ways = $16 - 2 = 14$. 3. Problem 19: Divide 4 distinct fruits into 2 identical baskets, no basket empty. Number of ways to partition 4 distinct items into 2 unlabeled nonempty subsets is the Stirling number of the second kind $S(4,2) = 7$. 4. Problem 20: Sarah gives 2 books to John and 1 poster to Mary from 5 distinct books and 3 distinct posters. Step 1: Choose 2 books for John: $\binom{5}{2} = 10$ ways. Step 2: Choose 1 poster for Mary: $\binom{3}{1} = 3$ ways. Total ways = $10 \times 3 = 30$. 5. Problem 21: Committees of 5 men and 6 women from 10 men and 12 women. Number of ways = $\binom{10}{5} \times \binom{12}{6}$. Calculate: $\binom{10}{5} = 252$, $\binom{12}{6} = 924$. Total = $252 \times 924 = 232848$. 6. Problem 22: Team of 5 from 4 girls and 7 boys with at least 3 girls. Cases: - 3 girls, 2 boys: $\binom{4}{3} \times \binom{7}{2} = 4 \times 21 = 84$ - 4 girls, 1 boy: $\binom{4}{4} \times \binom{7}{1} = 1 \times 7 = 7$ Total = $84 + 7 = 91$. 7. Problem 23: 5-letter words with 3 consonants and 2 vowels from 7 consonants and 4 vowels. Step 1: Choose 3 consonants: $\binom{7}{3} = 35$. Step 2: Choose 2 vowels: $\binom{4}{2} = 6$. Step 3: Arrange 5 letters: $5! = 120$. Total = $35 \times 6 \times 120 = 25200$. 8. Problem 24: Choose 2 pants, 3 shirts, 1 sweater from 4 pants, 7 shirts, 3 sweaters. Number of ways = $\binom{4}{2} \times \binom{7}{3} \times \binom{3}{1}$. Calculate: $\binom{4}{2} = 6$, $\binom{7}{3} = 35$, $\binom{3}{1} = 3$. Total = $6 \times 35 \times 3 = 630$. 9. Problem 25: Choose 5 movies with at least 2 Westerns from 8 Western and 12 Sci-Fi. Cases: - 2W, 3SF: $\binom{8}{2} \times \binom{12}{3} = 28 \times 220 = 6160$ - 3W, 2SF: $\binom{8}{3} \times \binom{12}{2} = 56 \times 66 = 3696$ - 4W, 1SF: $\binom{8}{4} \times \binom{12}{1} = 70 \times 12 = 840$ - 5W, 0SF: $\binom{8}{5} = 56$ Total = $6160 + 3696 + 840 + 56 = 10752$.