A signal can be sent from one location to another by running different colored flags up a flagpole, one above the other. There are 15 different colored flags to choose from but only 5 flags will be flown. Find the number of different signals consisting of 5, if the first flag must be red.

If the first flag is red then the number of different signals consisting of 5 flags is

To solve the problem, we note the following:

1. There are **15 different colored flags**, one of which is **red**.
2. The **first flag must be red**, so this position is fixed.
3. The remaining 4 flags can be chosen from the other **14 colors** (since red is already used).

### Step 1: Choose the remaining 4 flags
We can select the 4 flags from the remaining 14 colors. The number of ways to arrange 4 flags out of 14 in a specific order is calculated using the **permutation formula**:

\[
P(n, r) = \frac{n!}{(n-r)!}
\]

Here, \(n = 14\) and \(r = 4\). Thus:

\[
P(14, 4) = \frac{14!}{(14-4)!} = \frac{14 \times 13 \times 12 \times 11}{1} = 24,024
\]

### Step 2: Multiply by the fixed first flag
Since the first flag is fixed as red, it doesn't change the number of permutations for the remaining 4 flags. Hence, the total number of signals is the same as the number of arrangements of the 4 remaining flags:

\[
\text{Total number of signals} = 24,024
\]

Thus, **if the first flag is red, the number of different signals consisting of 5 flags is 24,024**.

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Do you need more details or have questions?  
Here are five related questions to explore further:  

1. How would the solution change if the first flag wasn't required to be red?  
2. How does the permutation formula differ from a combination formula in such problems?  
3. What if only 3 out of the 5 flags had to be distinct colors?  
4. How can the solution be adapted if we include flags of the same color but in different positions?  
5. What is the general formula for signals when the first \( k \) flags are fixed?  

**Tip:** Fixing an element in a sequence simplifies counting by reducing the number of available options.

A signal can be sent from one location to another by running different colored flags up a flagpole, one above the other. There are 15 different colored flags to choose from but only 5 flags will be flown. Find the number of different signals consisting of 5, if the first flag must be red.

Learn how to find the number of unique 5-flag signals that can be formed using 15 different colors, given that the first flag must be red. This detailed solution explores permutations and counting principles, ideal for Grades 10-12.

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