Permutations Quant Problem

GRE Quant Problem Solving
1

Hi, I have a query regarding this problem:

“In a high school talent show, there are 5 different acts to be arranged in a lineup. Two acts only involve singing, two only involve dancing, and one only involves comedy routines. To ensure variety in the show, the organizers want to arrange the acts so that no two acts of the same type (singing, dancing, or comedy) are performed consecutively. How many different possible arrangements meet this requirement, assuming the order of acts of the same type does not matter?”

I’m unable to understand where I’m going wrong in solving this.

My method:
We calculate the total number of permutations possible, which would be 5! since there are 5 entities in total (S1,S2, D1, D2, C).
5! = 120

Then we calculate the number of permutations where acts of the same type occur together (order doesn’t matter). Since there would be 3 entities we’d consider, the number of permutations would be 3! = 6

The total number of permutations - the total number of permutations where acts of same type occur together = number of permutations where acts of the same type don’t occur together = 114.

I’m unable to understand where I’m going wrong with this.
Could someone please help me?

2

What about (S1, S2, D1, C, D2)? According to you this works (because you don’t subtract such cases away), but the question suggests otherwise.

3

Oh, that’s interesting. I didn’t think of those cases before. Thanks!
What do you think would be the best strategy to solve this?
Brute force and write each case? Or is there a way to do this using the permutations formula?