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Answer :
Final answer:
The number of different signals consisting of 4 flags, with the first flag being orange, is 286.
Explanation:
The problem asks us to determine the number of different signals consisting of 4 flags if the first flag must be orange. Since the first flag is already determined, we need to choose the remaining 3 flags from the 14 remaining coloured flags. This can be done using combinations. There are 14 coloured flags to choose from, and we need to choose 3 of them. The number of different signals would then be 14 choose 3 or C(14,3).
The formula for combinations is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of objects to choose from and r is the number of objects to choose. In this case, n = 14 and r = 3.
Substituting the values into the formula, we have:
C(14,3) = 14! / (3!(14-3)!) = 14! / (3! * 11!) = (14 * 13 * 12) / (3 * 2 * 1) = 286
Therefore, the number of different signals consisting of 4 flags, with the first flag being orange, is 286.
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