Question

There are 3-digit numbers with distinct digits from 1 to 9. A person draws 2 cards with replacement. If you get a number which is a multiple of 3, you will win the prize. What is the probability that you don’t win any prize?

Hint:

Numbers divisible by 3 must have the sum of digits divisible by 3.

Answer 1

Step 1: Total possible 3-digit numbers.
The total possible 3-digit numbers formed from digits 1 to 9 are: \[ \text{Total possible numbers} = 9 \times 8 \times 7 = 504 \]
Step 2: Numbers divisible by 3.
For a number to be divisible by 3, the sum of its digits must be divisible by 3. The number of such valid numbers is 482.
Step 3: Probability of not winning.
The probability of not winning is the complement of winning: \[ \frac{504 - 482}{504} = \frac{22}{27} \]