Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce, 1 poplar, 1 willow, and 1 elm.Assuming that the 4 birches are completely identical, how many possible arrangements of trees are there?a.40,320b.8c.2,480d.1,680
Question
Bart wants to plant 8 trees in a row along his fence. He has been given 4 birches, 1 spruce, 1 poplar, 1 willow, and 1 elm.Assuming that the 4 birches are completely identical, how many possible arrangements of trees are there?a.40,320b.8c.2,480d.1,680
Solution
The problem is asking for the number of ways to arrange 8 trees, where 4 of them are identical (birches).
This is a permutation problem with repetition. The formula for permutations with repetition is n! / (n1! * n2! * ... * nk!), where n is the total number of items, and n1, n2, ..., nk are the numbers of identical items.
In this case, n = 8 (total number of trees), n1 = 4 (number of birches), and n2 = n3 = n4 = n5 = 1 (for the spruce, poplar, willow, and elm, respectively).
So, the number of arrangements is 8! / (4! * 1! * 1! * 1! * 1!) = 40,320 / 24 = 1,680.
So, the correct answer is d. 1,680.
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