Mike, Mo, and Jo want to sit together, in any order, to watch the basketball game with their 3 other friends. The other friends don't care where they sit. How many ways can this group sit along a bench?
I did: (3!)(3!)(4)=144. Would this be correct?
Yes, this is correct.
No, your calculation is not correct.
To find the total number of ways the group can sit together, we can treat Mike, Mo, and Jo as a single entity, since they want to sit together and the order in which they sit does not matter. This group of three can be arranged among themselves in 3! = 6 ways (since they can be seated in any order).
Now, we have a total of 4 + 1 = 5 entities (the group of three and the three other friends) to be seated in a row. The number of ways these 5 entities can be arranged is 5!, which is equal to 5 x 4 x 3 x 2 x 1 = 120.
Therefore, the correct answer is (3!)(5!) = (6)(120) = 720. There are 720 different ways for this group to sit along the bench.
To determine the number of ways the group can sit along the bench, we need to consider the seating arrangement of the three friends who want to sit together (Mike, Mo, and Jo), as well as the seating arrangement of the other three friends who don't care where they sit.
First, let's consider the three friends who want to sit together. Since they can sit in any order, we can calculate the number of seating arrangements for them by taking the factorial of 3, denoted as 3!.
Next, let's consider the three friends who don't care where they sit. Since they can also sit in any order, we can calculate the number of seating arrangements for them by taking the factorial of 3, denoted as 3!.
Finally, we multiply the number of seating arrangements for the three friends who want to sit together by the number of seating arrangements for the three friends who don't care where they sit, as they are independent of each other.
Therefore, the total number of seating arrangements is given by (3!) * (3!) = 6 * 6 = 36.
Hence, your previous calculation of (3!) * (3!) * 4 = 144 is incorrect. The correct answer is 36.