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 9 flags that can be made using 3 white flags, 3 red flags, and 3 blue flags.

Question

25-04-2025
Mathematics

College

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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 9 flags that can be made using 3 white flags, 3 red flags, and 3 blue flags.

Answer :

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psm22415 psm22415 · Answer 1 · 2025-03-29T10:47:32-04:00

The total number of different signals consisting of flags is 216

Given data ,

The number of different signals consisting of 9 flags that can be made using 3 white flags, 3 red flags, and 3 blue flags can be calculated using combinatorial principles.

There are 3 choices for the top flag (white, red, or blue), 2 choices for the second flag (since we've already used one color for the top flag), and 1 choice for the third flag.

Similarly, there are 3 choices for the fourth flag, 2 choices for the fifth flag, and 1 choice for the sixth flag. Finally, there are 3 choices for the seventh flag, 2 choices for the eighth flag, and 1 choice for the ninth flag.

Using the multiplication principle, we can multiply the number of choices at each step to get the total number of different signals:

3 × 2 × 1 × 3 × 2 × 1 × 3 × 2 × 1 = 3! × 3! × 3!

where "!" represents the factorial operation, which is the product of all positive integers up to a given number.

So, the total number of different signals consisting of 9 flags that can be made using 3 white flags, 3 red flags, and 3 blue flags is:

3! × 3! × 3! = 6 × 6 × 6 = 216

Hence , the number of signals is 216

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