This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Arrange the boys. There are 7 boys. The number of ways to arrange these 7 boys on a bench is 7!. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 Step 2: Identify the possible positions for the girls. When the 7 boys are arranged, they create 8 possible spaces (gaps) where the girls can sit so that no two girls are next to each other. Let 'B' represent a boy and '_' represent a possible position for a girl: \_ B \_ B \_ B \_ B \_ B \_ B \_ B \_ There are 8 such positions. Step 3: Place the girls in the available positions. We have 2 girls to place in 8 distinct positions. Since the girls are distinct, the order in which they are placed matters. This is a permutation problem. The number of ways to choose 2 positions out of 8 and arrange the 2 girls in them is given by P(8, 2). P(8, 2) = (8!)/((8-2)!) = (8!)/(6!) = 8 × 7 = 56 Step 4: Calculate the total number of arrangements. To find the total number of ways, multiply the number of ways to arrange the boys by the number of ways to place the girls. Total ways = (Ways to arrange boys) × (Ways to place girls) Total ways = 7! × P(8, 2) Total ways = 5,040 × 56 Total ways = 282,240 The final answer is 282,240.