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Answer :
There are 1365 different signals consisting of 5 flags, where the first flag must be white. To find the number of different signals consisting of 5 and if the first flag must be white, we can use the concept of combinations.
Since the first flag must be white, we have one fixed choice. Now, we need to choose the remaining 4 flags from the 15 different colored flags available.
The number of ways to choose k items from a set of n items is given by the combination formula:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
where [tex]\(n!\)[/tex] represents the factorial of n, which is the product of all positive integers from 1 to n.
In this case, we want to find [tex]\(C(15, 4)\)[/tex] since we need to choose 4 flags from the remaining 15 colors (excluding white, which is already chosen).
Substituting the values into the combination formula:
[tex]\[ C(15, 4) = \frac{15!}{4!(15-4)!} \][/tex]
Now, we need to calculate the factorials:
[tex]\[ 15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]\[ 4! = 4 \times 3 \times 2 \times 1 \]\[ 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
Now, plug the values back into the combination formula:
[tex]\[ C(15, 4) = \frac{15!}{4!(15-4)!} = \frac{15!}{4! \times 11!} \][/tex]
Now, cancel out the common terms:
[tex]\[ C(15, 4) = \frac{15 \times 14 \times 13 \times 12}{4 \times 3 \times 2 \times 1} \]\[ C(15, 4) = 1365 \][/tex]
So, there are 1365 different signals where the first flag must be white.
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