Questions: A signal can be formed by running different colored flags up a pole, one above the other. Find the number of different signals consisting of 7 flags that can be made using 3 white flags, 2 red flags, and 2 blue flags.

A signal can be formed by running different colored flags up a pole, one above the other. Find the number of different signals consisting of 7 flags that can be made using 3 white flags, 2 red flags, and 2 blue flags.
Transcript text: A signal can be formed by running different colored flags up a pole, one above the other. Find the number of different signals consisting of 7 flags that can be made using 3 white flags, 2 red flags, and 2 blue flags.
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Solution

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Solution Steps

Step 1: Calculate the factorial of the total number of flags (N!)

Given a total of 7 flags, \(N! = 5040\).

Step 2: Calculate the product of the factorials of the number of flags of each color

For the given numbers of flags of each color [3, 2, 2], the product of their factorials is \( (3!), (2!), (2!) = 24 \).

Step 3: Calculate the number of different signals using the formula

Using the formula \( P = \frac{N!}{n_1! \cdot n_2! \cdot ... \cdot n_k!} \), we find \( P = \frac{5040}{24} = 210 \) different signals.

Final Answer:

The number of different signals that can be formed is 210.

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