SOLUTION: In how many ways can 8 chairs be arranged in a row if there are 3 blue chairs, 4 red chairs and 1 balck chair? Show your solution

I will assume that the chairs of the same color are indistinguishable.

Of the 8 chair positions we can pick 3 of them to put the blue chairs
in.  That's (8C3) places to put the 3 blue chairs in.

Of the remaining 5 chair positions, we can pick 4 of them to place the
4 red chairs.  That's (5C4) chair positions to put the red chairs in.
 
Then we place the 1 black chair in the remaining (1C1) = 1 position.

Answer (8C3)(5C4)(1C1) = (56)(5)(1) = 280 ways.
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By the way, if we could tell all the chairs of the same color apart, we
would simply use permutations instead of combinations.
(8P3)(5P4)(1P1) = (336)(120)(1) = 40320 ways.

Edwin