combination of two sets

Question

I have set of two baskets one with apples other with oranges. Total number of apples and oranges are not known. Problem is to choose 4 apples and oranges and arrange them in a line, such that maximum of two apples are close to each other and also not more than three oranges are close to each other, with this constraint how many sets you we make.

Answer 1

Assume you mean 4 fruits (all possible mixtures of apples and oranges) with the given constraints for arranging in a line. Four identical fruits disobey the constraints.

(1) 3 apples and an orange: 

{ AAOA, AOAA, OAAA }

(2) 2 apples and 2 oranges: 

{ AAOO, AOAO, AOOA, OAAO, OAOA, OOAA }

(3) 1 apple and 3 oranges:

{ AOOO, OAOO, OOAO, OOOA }

Total arrangements from the sets=13.