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Answer :
There are 6435 distinguishable arrangements using all of the flags, satisfying the given conditions.
To determine the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent, we can use the concept of permutations with restrictions.
Let's consider the flagpoles as two separate entities. We need to distribute the 10 blue flags and 9 green flags between the two flagpoles in a way that satisfies the given conditions.
First, let's place one blue flag on each flagpole, which leaves us with 8 blue flags and 9 green flags to distribute.
To ensure that no two green flags are adjacent on either pole, we can imagine arranging the green flags in a line with spaces between them. There are 8 spaces between the 9 green flags. We need to distribute the blue flags among these spaces.
To do this, we can use the concept of stars and bars. We have 8 blue flags to distribute among the 8 spaces, so we can represent this as placing 8 stars (blue flags) in 8 spaces (bars). The number of arrangements can be calculated using the formula (n+r-1)C(r-1), where n is the number of stars (8) and r is the number of bars (8).
Using this formula, we get (8+8-1)C(8-1) = 15C7 = 6435.
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