Question 8: How many ways can you arrange the letters of the word "COMBINATION"?

Here are the solutions to the problems: Question 8: How many ways can you arrange the letters of the word "COMBINATION"? The word "COMBINATION" has 11 letters. Count the occurrences of each letter: C: 1 O: 2 M: 1 B: 1 I: 2 N: 2 A: 1 T: 1 This is a permutation with repetitions. The formula is (n!)/(n_1! n_2! ... n_k!), where n is the total number of letters and n_i are the counts of repeated letters. Step 1: Identify the total number of letters and the counts of repeated letters. Total letters n = 11. Repeated letters: O (2 times), I (2 times), N (2 times). Step 2: Apply the permutation with repetitions formula. (11!)/(2! × 2! × 2!) = (39916800)/(2 × 2 × 2) = (39916800)/(8) = 4989600 The calculated answer is 4,989,600. This is not among options a), b), c), or d). The final answer is None of the above. Question 9: In how many ways can you arrange the letters of the word "SUCCESS"? The word "SUCCESS" has 7 letters. Count the occurrences of each letter: S: 3 U: 1 C: 2 E: 1 This is a permutation with repetitions. Step 1: Identify the total number of letters and the counts of repeated letters. Total letters n = 7. Repeated letters: S (3 times), C (2 times). Step 2: Apply the permutation with repetitions formula. (7!)/(3! × 2!) = (5040)/((3 × 2 × 1) × (2 × 1)) = (5040)/(6 × 2) = (5040)/(12) = 420 The calculated answer is 420. This is not among options a), b), c), or d). The final answer is None of the above. Question 10: If there are 5 men and 4 women, how many ways can a committee of 3 men and 2 women be formed? This is a combination problem, as the order of selection for committee members does not matter. We need to choose 3 men from 5 men AND 2 women from 4 women. Step 1: Calculate the number of ways to choose 3 men from 5. 53 = (5!)/(3!(5-3)!) = (5!)/(3!2!) = (5 × 4)/(2 × 1) = 10 Step 2: Calculate the number of ways to choose 2 women from 4. 42 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4 × 3)/(2 × 1) = 6 Step 3: Multiply the results from Step 1 and Step 2 to find the total number of ways to form the committee. 10 × 6 = 60 The calculated answer is 60. This is not among options a), b), c), or d). The final answer is None of the above. Question 11: A bag contains 6 red balls and 4 green balls. In how many ways can 3 balls be drawn without replacement? Total number of balls = 6 (red) + 4 (green) = 10 balls. We need to choose 3 balls from these 10. Since the order of drawing does not matter, this is a combination problem. Step 1: Use the combination formula nk = (n!)/(k!(n-k)!). Here, n=10 and k=3. 103 = (10!)/(3!(10-3)!) = (10!)/(3!7!) Step 2: Calculate the value. 103 = (10 × 9 × 8)/(3 × 2 × 1) = 10 × 3 × 4 = 120 The correct option is b). The final answer is 120. Question 12: How many different