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Answer :
To find the number of distinct arrangements of 8 flags hung in a vertical line consisting of 4 distinct red flags, 3 distinct white flags, and 1 blue flag, use the permutations of a multiset formula resulting in 5,040 different arrangements.
This is a permutation problem with repetition, which falls under the subject of combinatorics in mathematics. To solve this problem, we can use the formula for permutations of a multiset: n! / (p1! imes p2! imes ... times pk!), where n is the total number of items to arrange, and p1, p2, ..., pk are the frequencies of each distinct item.
In this particular problem, n = 8 (the total number of flags), and we have 3 distinct groups of flags: 4 red (p1), 3 white (p2), and 1 blue (p3). Plugging into the formula, we get 8! / (4! imes 3! imes 1!) = 5,040 distinct arrangements.
Therefore, there are 5,040 different ways to hang the flags in a vertical line.
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