2002 Indonesia MO Problems/Problem 2
Problem
Five regular dices are thrown, one at each time, then the product of the numbers shown are calculated. Which probability is bigger; the product is or the product is ?
Solution
Let be the roll of the first dice, be the roll of the second dice, be the roll of the third dice, be the roll of the fourth dice, and be the roll of the fifth dice. To calculate which probability is bigger, find the number of ways to roll dice that result in the two wanted values. Note that the prime factorization of is and the prime factorization of is .
- If the product of the five dices is , then , where . To find the number of ways, create casework based on the number of ones.
- For the case of no , the only way that works is , for a total of possibilities.
- For the case of one , the two ways that work are and , for a total of possibilities.
- For the case of two , the only way that works is , for a total of possibilities.
- If the product of the five dices is , then , where . To find the number of ways, create casework based on the number of ones.
- For the case of no , the two ways that work are and , for a total of possibilities.
- For the case of one , the three ways that work are and and , for a total of possibilities.
- For the case of two , the only way that works is , for a total of possibilities.
Tallying up the results yields ways to get and ways to get , so the bigger probability is the product being .
See Also
2002 Indonesia MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 3 |
All Indonesia MO Problems and Solutions |