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 ___

Modern Math

Permutation & Combination

Hard

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

Entered answer:

✅ Correct Answer: 54

Total ways to choose $3$ distinct integers from $9$ :

$\binom{9}{3} = 84$

It is easier to use the complement: count triples whose product is NOT divisible by $4$ , then subtract from $84$ .

Classify $\{1, 2, \ldots, 9\}$ by power of $2$ :

Odd (no factor of $2$ ): $\{1, 3, 5, 7, 9\}$ , count $= 5$ .

Exactly one factor of $2$ : $\{2, 6\}$ , count $= 2$ .

At least two factors of $2$ : $\{4, 8\}$ , count $= 2$ .

The product is not divisible by $4$ when the total power of $2$ in the triple is $0$ or $1$ :

All three odd: $\binom{5}{3} = 10$ .

One number from $\{2, 6\}$ and two odd: $\binom{2}{1} \times \binom{5}{2} = 2 \times 10 = 20$ .

Total not divisible by $4 = 10 + 20 = 30$ .

Number of triples with product divisible by $4 = 84 - 30 = 54$ .

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