AMC 10 Daily Practice Round 2

Observe that for each such arrangement, we can remove apples between each of the pairs of adjacent orange (i.e. pairs of orange with no orange between them) to obtain an arbitrary arrangement of apples and oranges. Conversely, starting with any arrangement of apples and oranges, adding apples between each pair of adjacent oranges yields an arrangement of apples and oranges such that any oranges are separated by at least apples. Hence, the number of desired arrangements equals the number of arrangements of apples and oranges, or .