Given 4 flags of different colours, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?
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Step-by-Step Solution
Step 1: Identify the problem type
This problem involves selecting and arranging a specific number of items from a larger set, where the order of selection matters. This is a classic permutation problem because placing a red flag above a blue flag is a different signal than placing a blue flag above a red flag.
Step 2: Determine the total number of items and items to be arranged
We have 4 flags of different colors, so the total number of items available is n=4. A signal requires the use of 2 flags, so the number of items to be arranged is r=2.
Step 3: Apply the permutation formula
The formula for permutations of n items taken r at a time is P(n,r)=(n−r)n!!. This formula calculates the number of ways to arrange r items from a set of n distinct items.
Step 4: Substitute values and calculate
Substitute n=4 and r=2 into the permutation formula. Calculate the factorials and simplify the expression to find the total number of different signals.