In how many ways can 3 boys and 2 girls line up for a group picture? In how many ways can they line up if a boy is to be at each end?
Well, let's see here. To line up for a group picture, we have a total of 5 people - 3 boys and 2 girls. So, for the first part of the question, we can use the concept of permutations. We simply multiply the number of possible choices for each position.
For the 5 people, the first position can be filled by any of the 5 people. Once the first position is filled, the second position can be filled by any of the remaining 4 people. Similarly, the third position can be filled by any of the remaining 3 people. For the fourth position, we have 2 people left, and finally, for the fifth position, only 1 person remains.
So, the total number of ways to line up 3 boys and 2 girls is simply:
5 x 4 x 3 x 2 x 1 = 120
Now, for the second part of the question, we need to have a boy at each end. This means that the first and the last positions must be boys.
For the first position, we have 3 boys to choose from. Once the first position is filled by a boy, we are left with 4 people to fill the remaining 4 positions. For the second position, we have 4 choices left, then 3 choices for the third position, 2 choices for the fourth position, and finally 1 choice for the fifth position.
So, the number of ways to line up with a boy at each end is:
3 x 4 x 3 x 2 x 1 = 72
I hope that answered your question, and boy oh boy, weren't those some interesting calculations!
To find the number of ways the 3 boys and 2 girls can line up for a group picture, we can calculate the different permutations.
Step 1: Find the total number of arrangements for all 5 people.
Since there are 5 people, we calculate the number of arrangements using the formula n!, where n is the number of people.
Therefore, the total number of arrangements for all 5 people is 5!
5! = 5 x 4 x 3 x 2 x 1 = 120
So, there are 120 possible arrangements for all 5 people.
Step 2: Find the number of arrangements if a boy is to be at each end.
Since there are 3 boys and 2 girls, we can think of the boys as one group, and the girls as another group.
For the boys, there are 3! (3 factorial) ways to arrange them amongst themselves.
Similarly, for the girls, there are 2! (2 factorial) ways to arrange them amongst themselves.
Considering the boys and girls as separate groups, we can multiply the number of arrangements for each group together to find the total number of arrangements.
3! x 2! = 6 x 2 = 12
So, there are 12 possible arrangements if a boy is to be at each end.
Therefore, the answer to the question is:
- In total, there are 120 possible ways for the 3 boys and 2 girls to line up for a group picture.
- If a boy is to be at each end, there are 12 possible ways for them to line up.
To find the number of ways the boys and girls can line up for a group picture, we can use the concept of permutations.
1. For the first question, where there are 3 boys and 2 girls, we need to calculate the total number of permutations. This can be done by multiplying the number of ways to arrange the boys with the number of ways to arrange the girls.
Number of ways to arrange the boys = 3! (3 factorial) = 3 * 2 * 1 = 6
Number of ways to arrange the girls = 2! (2 factorial) = 2 * 1 = 2
Total number of permutations = Number of ways to arrange boys * Number of ways to arrange girls = 6 * 2 = 12
Therefore, there are 12 ways for 3 boys and 2 girls to line up for a group picture.
2. For the second question, where a boy is to be at each end, we can fix the positions for the boys and then calculate the number of ways to arrange the remaining people (girls and the remaining boy).
Number of ways to arrange the girls and the remaining boy = 2! (2 factorial) = 2 * 1 = 2
Therefore, there are 2 ways for the 3 boys and 2 girls to line up if a boy is to be at each end.
I hope this helps to clarify the process of finding the answers! Let me know if you have any further questions.