How many signals can be made by hoisting 5 flags of different colours ?

Answer

To find the total number of signals that can be made by hoisting 5 flags of different colors, we can consider the number of ways to hoist 1 flag, 2 flags, 3 flags, 4 flags, and all 5 flags. We will use the concept of permutations since the order in which the flags are hoisted matters.

### Step-by-step Solution:

1. **Hoisting 5 Flags**:
- The number of ways to hoist all 5 flags is given by the permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \).
- Here, \( n = 5 \) and \( r = 5 \).
- So, \( P(5, 5) = \frac{5!}{(5-5)!} = \frac{5!}{0!} = 5! = 120 \).

2. **Hoisting 4 Flags**:
- The number of ways to hoist 4 flags out of 5 is \( P(5, 4) \).
- So, \( P(5, 4) = \frac{5!}{(5-4)!} = \frac{5!}{1!} = \frac{120}{1} = 120 \).

3. **Hoisting 3 Flags**:
- The number of ways to hoist 3 flags out of 5 is \( P(5, 3) \).
- So, \( P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 \).

4. **Hoisting 2 Flags**:
- The number of ways to hoist 2 flags out of 5 is \( P(5, 2) \).
- So, \( P(5, 2) = \frac{5!}{(5-2)!} = \frac{5!}{3!} = \frac{120}{6} = 20 \).

5. **Hoisting 1 Flag**:
- The number of ways to hoist 1 flag out of 5 is \( P(5, 1) \).
- So, \( P(5, 1) = \frac{5!}{(5-1)!} = \frac{5!}{4!} = \frac{120}{24} = 5 \).

6. **Total Signals**:
- Now, we sum all the possibilities:
\[ \text{Total Signals} = P(5, 5) + P(5, 4) + P(5, 3) + P(5, 2) + P(5, 1) = 120 + 120 + 60 + 20 + 5 = 325 \]

### Final Answer:
The total number of signals that can be made by hoisting 5 flags of different colors is **325**.