Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different flags are available.
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Step-by-Step Solution
Step 1: Understand the Problem
We are given that there are N=5 different flags available. We need to find the total number of signals that can be generated by arranging at least 2 flags. This means we can use 2 flags, 3 flags, 4 flags, or all 5 flags.
Step 2: Calculate arrangements for 2 flags
When arranging 2 flags out of 5 different flags, the order matters, so we use permutations. The number of permutations of n items taken r at a time is given by P(n,r)=(n−r)n!!. For 2 flags, this is P(5,2).
Step 3: Calculate arrangements for 3 flags
Similarly, for arranging 3 flags out of 5, we calculate P(5,3). This involves selecting 3 flags and arranging them in order.
Step 4: Calculate arrangements for 4 flags
For arranging 4 flags out of 5, we calculate P(5,4). This means selecting 4 flags and arranging them in a specific order.
Step 5: Calculate arrangements for 5 flags
Finally, for arranging all 5 flags out of 5, we calculate P(5,5). Remember that 0!=1.
Step 6: Sum the arrangements
To find the total number of different signals, we sum the number of arrangements for 2, 3, 4, and 5 flags, as the problem states 'at least 2 flags'.