The number of ways in which three distinct integers can be chosen from the set {1, 2, …, 9} such that their product is divisible by 4, is_________

Choose 3 distinct integers from $$\{1,2,\ldots,9\}$$ such that the product is divisible by 4.

Total triples: $$\binom{9}{3} = 84$$.

Subtract triples whose product is NOT divisible by 4: triples with at most one factor of 2 in the product.

• All three odd: $$\binom{5}{3} = 10$$.
• Exactly one number from $$\{2,6\}$$ (each contributing exactly one factor of 2) and two odd: $$2 \cdot \binom{5}{2} = 20$$.

Not-div-by-4 = $$10 + 20 = 30$$. Divisible by 4 = $$84 - 30 = 54$$.