Number of distinct flag signals using 5 different coloured flags when any non empty selection and order is allowed

Introduction / Context:
This question models signal making with different coloured flags. Each signal is formed by hoisting some flags in a particular order. The key idea is that any number of flags from one to all five may be used, and no flag is repeated within one signal.

Given Data / Assumptions:

• There are 5 flags, each of a different colour.
• Any number of flags can be hoisted at a time, from 1 to 5.
• Within a signal, flags are arranged in order, so sequence matters.
• No flag is repeated in a single signal.
• Signals with different orders or different chosen flags are considered distinct.

Concept / Approach:
For a fixed number r of flags, the number of ordered arrangements of r flags chosen from 5 distinct flags is given by permutations, written as 5Pr. Since r can be 1, 2, 3, 4, or 5, we sum these permutation counts over all possible r.

Step-by-Step Solution:
Step 1: For r = 1, the number of signals is 5P1 = 5.Step 2: For r = 2, the number of signals is 5P2 = 5 * 4 = 20.Step 3: For r = 3, the number of signals is 5P3 = 5 * 4 * 3 = 60.Step 4: For r = 4, the number of signals is 5P4 = 5 * 4 * 3 * 2 = 120.Step 5: For r = 5, the number of signals is 5P5 = 5! = 120.Step 6: Sum all signals: 5 + 20 + 60 + 120 + 120 = 325.

Verification / Alternative check:
Notice that 5P1 through 5P5 are standard permutation values and the sum 325 is reasonable given the rapid growth of permutations. There is no simpler closed form for this sum here, so direct addition is the efficient and accurate method.

Why Other Options Are Wrong:

• 235 and 253 are smaller than 325 and may result from omitting some terms, for example summing only up to 5P4.
• None of these is incorrect because option 325 exactly matches the correct total.

Common Pitfalls:
Students sometimes confuse this with combination counting and use 2^5 minus 1, which counts unordered selections, not signals where order matters. Another mistake is assuming only one flag can be hoisted, leading to a count of 5. Always check whether the order of flags in a signal changes its meaning.

Final Answer:
The number of distinct flag signals that can be generated is 325, so the correct answer is 325.