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
Welcome back Immanuel — been a while! Let's pick up where you left off. The problem asks for the number of ways 7 boys and 2 girls can sit on a bench so that the girls do not sit next to each other. Step 1: Calculate the total number of ways to arrange all 9 people without any restrictions. There are 7 boys and 2 girls, making a total of 7+2=9 people. The total number of ways to arrange 9 distinct people is 9!. 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880 Step 2: Calculate the number of ways the 2 girls sit together. To find the arrangements where the two girls sit together, treat the two girls as a single unit. Now we have 7 boys and 1 unit of girls, making a total of 7+1=8 units to arrange. The number of ways to arrange these 8 units is 8!. 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 Within the unit of the two girls, the girls can arrange themselves in 2! ways (Girl 1 then Girl 2, or Girl 2 then Girl 1). 2! = 2 × 1 = 2 So, the total number of ways the two girls sit together is the product of these two numbers: 8! × 2! = 40,320 × 2 = 80,640 Step 3: Subtract the arrangements where the girls sit together from the total arrangements. The number of ways the girls do not sit next to each other is the total number of arrangements minus the number of arrangements where they do sit next to each other. 362,880 - 80,640 = 282,240 The number of ways they can arrange themselves so that the girls do not sit next to each other is 282,240. Send me the next one 📸