How many ordered triples of positive integers satisfy ABC equals 4000? - Answers

To find the ordered triples of positive integers ( (A, B, C) ) such that ( ABC = 4000 ), we first factor 4000 into its prime factors: ( 4000 = 2^5 \times 5^3 ). The number of ways to distribute the prime factors among ( A ), ( B ), and ( C ) can be calculated using the stars and bars combinatorial method. For the power of 2, we have ( 5 + 3 - 1 = 7 ) positions to place the dividers, yielding ( \binom{7}{2} = 21 ) ways. For the power of 5, we have ( 3 + 3 - 1 = 5 ) positions, yielding ( \binom{5}{2} = 10 ) ways. Multiplying these results gives ( 21 \times 10 = 210 ) ordered triples ( (A, B, C) ).