A troupe of 12 dancers is going to line up on stage. Of these dancers, 6 are wearing black and 6 are wearing red. A dancer wearing black must be in the first position and they must alternate between colors. In how many ways can the dancers line up?
Question
A troupe of 12 dancers is going to line up on stage. Of these dancers, 6 are wearing black and 6 are wearing red. A dancer wearing black must be in the first position and they must alternate between colors. In how many ways can the dancers line up?
Solution
To solve this problem, we need to consider the restrictions given.
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The first position must be a dancer in black. This means we have 6 choices for the first position.
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After the first position, the dancers must alternate in color. This means the second position must be a dancer in red. We have 6 choices for this position.
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For the third position, we must choose a dancer in black. Since we've already chosen one black dancer, we now have 5 choices.
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We continue this pattern, alternating between black and red dancers, with the number of choices decreasing by 1 each time.
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The total number of ways we can arrange the dancers is therefore 6 * 6 * 5 * 5 * 4 * 4 * 3 * 3 * 2 * 2 * 1 * 1 = 518,400 ways.
So, there are 518,400 ways the dancers can line up on stage.
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