Question

A troupe of 10 dancers is going to line up on stage. Of these dancers, 5 are wearing black and 5 are wearing red. A dancer wearing black must be in the first position and they must altemate between colors. In how many ways can the doncers line up?

Answer 1

There are 14,400 ways the dancers can line up on stage with the restriction that the first dancer must wear black and colors must alternate.

Step-by-step explanation:

The question refers to a permutation problem with restrictions. Since one dancer wearing black must be in the first position and the colors must alternate, starting with black, we can sequence the dancers by choosing from the remaining four black-clad dancers for the second black position, then three for the next, and so on. The red-clad dancers will follow similarly.

Calculation Steps

1. Select the first black dancer (5 choices since any of the 5 black dancers can be first).
2. Select the first red dancer (5 choices since after the first black dancer is chosen, any red can follow).
3. Choose the second black dancer (4 choices now).
4. Choose the second red dancer (4 choices after the first red is chosen).
5. Continue alternating selections for the remaining positions

Therefore, the total number of ways the dancers can line up is 5 x 5 x 4 x 4 x 3 x 3 x 2 x 2 x 1 x 1, which is equal to 14,400.

Answer 2

There are 14400 ways for the 10 dancers to line up on stage, alternating colors with the first dancer wearing black.

Step-by-step explanation:

The question involves determining the number of ways 10 dancers can line up on stage under certain conditions. Specifically, the dancers must alternate colors, and a dancer wearing black must be first.

Since there are 5 dancers of each color, and the dancers must alternate colors starting with a dancer in black, there are 5! (5 factorial) ways to arrange the dancers in black and 5! ways to arrange the dancers in red. However, since the first dancer must wear black, we don't have to consider multiple arrangements for the first position.

Therefore, the number of ways the dancers can line up is the product of the two separate arrangements:

5! × 5! = (5 × 4 × 3 × 2 × 1) × (5 × 4 × 3 × 2 × 1) = 120 × 120 = 14400.

So, there are 14400 possible ways for the dancers to line up on stage.