<p>How many different signals can be given using any number of flags from 4 flags of different colours?</p>
Step-by-Step Solution
Key Concept: Use permutations to count ordered arrangements when selecting different numbers of objects; sum all possibilities.
<p><strong>Solution:</strong></p><p>The signals can be made using one or more flags at a time. By the fundamental principle of addition, the total number of signals is:</p><p><span style="math">$^4P_1 + \,^4P_2 + \,^4P_3 + \,^4P_4$</span></p><p><span style="math">$= 4 + (4 \times 3) + (4 \times 3 \times 2) + (4 \times 3 \times 2 \times 1)$</span></p><p><span style="math">$= 4 + 12 + 24 + 24 = 64$</span></p>
Correct Answer: 64