Question

Three integers are chosen at random without replacement from the first 20 integers. The probability that their product is even is

• A. $7/19$
• B. $2/19$
• C. $17/19$
• D. $10/19$

Answer 1

The correct option is C $17/19$

The product of any number of integers will be even if and only if at least one of the integers involved is even. Let A denote the event that the product of the three integers is even. Then A' denotes the event that the product of the three integers is odd. The total number of ways of choosing three integers out of 1, 2, ..., 20 is ${}^{20}{C}_3$. The number of ways favourable to A' is ${}^{10}{C}_3$.
Therefore,$P\left(A^{\prime }\right)=\frac{{}^{10}{C}_3}{{}^{20}{C}_3}=\frac{10\times 9\times 8}{20\times 19\times 18}=\frac{2}{19}$
$\Rightarrow P\left(A\right)=1-P\left(A^{\prime }\right)=\frac{17}{19}$