Can someone tell me if I have the correct answers? I am really confused.
Three boys and three girls line up to go in the front door.
In how many ways can they line up?
My answer: 720
How many ways can they line up if the first one in line is a girl and then alternates by gender, a girl, a boy...
My answer: 3*3*3*2*2*1*1=108
Find the probabiltiy that the first person in line is a girl and the line alternates by gender.
My answer: 1/120
first, 6!, correct
second 3*3*2*2*1*1
third: 24/720*1/2
To determine if you have the correct answers, let's go through each question and explain how to calculate the answers.
1. In how many ways can they line up?
To find the number of ways they can line up, you need to calculate the number of possible permutations. Since there are 3 boys and 3 girls, there are 6 people in total. The number of ways to arrange 6 people in a line without any restrictions is 6!. However, since the boys and girls are indistinguishable within their groups, we need to divide by the factorial of the number of boys and girls separately (3! for each). So the number of ways they can line up is:
6! / (3! * 3!) = 720 / (6 * 6) = 720 / 36 = 20
So the correct answer is 20, not 720.
2. How many ways can they line up if the first one in line is a girl and then alternates by gender, a girl, a boy...
To calculate this, you start by placing the first girl in line (1 way), then choose one boy to stand after her (3 choices), then place the second girl (2 choices), followed by choosing one boy again (2 choices), and so on. So the total number of ways they can line up in this specific order is:
1 * 3 * 2 * 2 * 1 * 1 = 12
So the correct answer is 12, not 108.
3. Find the probability that the first person in line is a girl and the line alternates by gender.
To calculate the probability, you need to divide the number of favorable outcomes (the number of ways they can line up in the desired order) by the total number of possible outcomes (the total number of ways they can line up).
We already found that the number of ways to line up with the given conditions is 12. From the first question, we know that the total number of ways they can line up is 20. Therefore, the probability is:
12 / 20 = 0.6
So the correct answer is 0.6, not 1/120.