The formula for permutations of n items taken r at a time is P(n, r) = (n!)/((n-r)!). This formula calculates the number of ways to arrange r items from a set of n distinct items.

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?

Answer: 12 different signals can be generated.

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 $P(n, r) = \frac{n!}{(n-r)!}$ . 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.

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