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$\text{Give an apple to each child. You now have to distribute 3 apples and 6 oranges w/o any restrictions}\\ \text{This is a simple 2D stars and bars problem. You can independently distribute the apples and oranges}\\ N_A = \dbinom{3+4-1}{4-1} = \dbinom{6}{3} = 20\\ N_O = \dbinom{6+4-1}{4-1} = \dbinom{9}{3} = 84\\ N=N_A N_O = 1680$