Thus, the number of distinct eating sequences is $ \boxed{27720} $.Question: Suppose $ v $ is a positive multiple of $ 5 $. If $ v^3 < 1700 $, what is the greatest possible value of $ v $?

["Greatest Positive Multiple of 5 with $ v^3 < 1700 $", "When solving for the greatest positive multiple of 5 such that $ v^3 < 1700 $, we begin by identifying the cube root of 1700 to estimate the upper limit.", "We know:\n$$\n\sqrt[3]{1700} \approx 11.9\n$$\nThis means $ v $ must be less than or equal to 11.9. Since $ v $ is a positive multiple of 5, the possible values are:\n$$\nv = 5, 10\n$$\n(Note: $ 15^3 = 3375 $, which is much larger than 1700, so $ v = 15 $ is invalid.)", "Now compare the cubes:\n- $ 5^3 = 125 $\n- $ 10^3 = 1000 $\n- $ 15^3 = 3375 $ (too big)", "Among valid multiples of 5, $ v = 10 $ gives the largest cube under 1700.", "Thus, the greatest possible value of $ v $ is:\n$$\n\boxed{10}\n$$"]