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Answer :
The number of different signals consisting of 6 flags if the first flag must be blue is `15,120 different signals.`.
We have to find out the number of different signals consisting of 6 flags, if the first flag must be blue.If the first flag is blue, then there are 14 colors of flags that can be used to select the remaining 5 flags.
Using the fundamental principle of counting, the number of possible signals can be calculated by multiplying the number of options for each flag position.
That is:
14 options × 13 options × 12 options × 11 options × 10 options × 1 option= 240,240 different signals
However, this calculation also considers the order in which the flags are placed. The flags on the pole can be arranged in 6! ways. Since the order of the flags does not matter, each signal has been counted 6! times
.To get the number of different signals consisting of 6 flags, we need to divide 240,240 by 6! 240,240 / (6!)= 240,240 / 720= 333 different signals, to the nearest whole number.
The first flag is blue, hence, there are five more flags left to choose from the 14 remaining colors.
The number of signals that can be created with these five flags is: 13 × 12 × 11 × 10 × 9 = 15,120
The total number of signals that can be made using a blue flag as the first flag is:15,120 × 1 = 15,120 different signals.
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