The number of signals that can be generated by using 5 differently coloured flags, when any number of them may be hoisted at a time is :

Options

A. 235
B. 253
C. 325
D. none of these

Answer

To solve the problem of determining the number of signals that can be generated using 5 differently colored flags when any number of them may be hoisted at a time, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 differently colored flags and we can use any number of them (from 1 to 5) to create signals. Each arrangement of flags is considered a different signal.

2. **Using Permutations**: The number of ways to arrange \( r \) flags out of \( n \) flags is given by the formula \( P(n, r) = \frac{n!}{(n-r)!} \). Here, \( n = 5 \) (the total number of flags).

3. **Calculating for Each Possible Number of Flags**:
- For 1 flag:
\[ P(5, 1) = \frac{5!}{(5-1)!} = \frac{5!}{4!} = 5 \]
- For 2 flags:
\[ P(5, 2) = \frac{5!}{(5-2)!} = \frac{5!}{3!} = \frac{120}{6} = 20 \]
- For 3 flags:
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 \]
- For 4 flags:
\[ P(5, 4) = \frac{5!}{(5-4)!} = \frac{5!}{1!} = 120 \]
- For 5 flags:
\[ P(5, 5) = \frac{5!}{(5-5)!} = 5! = 120 \]

4. **Summing All Possible Arrangements**: Now, we add all the permutations calculated above:
\[ P(5, 1) + P(5, 2) + P(5, 3) + P(5, 4) + P(5, 5) = 5 + 20 + 60 + 120 + 120 \]
\[ = 325 \]

5. **Final Answer**: Therefore, the total number of signals that can be generated by using 5 differently colored flags is **325**.