Is the product odd?

Probability Level 3

If you take $3$ distinct natural numbers from $1$ to $100$ (inclusive), the probability that the product of the numbers is odd can be expressed as $\frac{A}{B}$ , where $A$ , $B$ both are coprime positive integers. What is the value of $A+B$ ?

3 4 9 37 None of the above

1 solution

Saad Khondoker
Mar 7, 2021

The product of a set of natural numbers will be odd if and only if all the selected numbers are odd.

There are $50$ odd numbers from $1$ to $100$ . The way to take $3$ from them is $^{50}C_{3}$ .

The way to take any $3$ numbers from $1$ to $100$ is $^{100}C_{3}$ .

So the probability is $\frac{^{50}C_{3}}{^{100}C_{3}}$ = $\frac{4}{33}$ .

So $A+B=4+33=37$ .

Is the product odd?

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