Rearranging Contestants?

📌 CONCEPT: The problem requires finding the total number of arrangements of all 20 dancers on the stage, given that the first group and the first 4 dancers of each subsequent group can be arranged in certain ways.

📐 RULE / FORMULA: The total number of arrangements can be found by multiplying the number of ways each group can be arranged. In this case, we have 5 groups, and the first group has 24 ways to arrange, while each of the subsequent groups has 24 ways to arrange the first 4 dancers. We need to use the multiplication principle to find the total number of arrangements.

💡 WORKED EXAMPLE: To find the total number of arrangements, we start by finding the number of ways the first group can be arranged, which is given as 24. For the remaining 4 groups, each group has 24 ways to arrange the first 4 dancers. We multiply these numbers together: 24 * 24 * 24 * 24 * 24 = 248,832,768. This is the total number of different arrangements of all 20 dancers.

⚠️ COMMON MISTAKE: Students may forget to multiply the number of arrangements for each group, leading to an incorrect calculation. It is essential to apply the multiplication principle to find the total number of arrangements.