Difference between revisions of "2002 Indonesia MO Problems/Problem 2"
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** For the case of no <math>1</math>, the two ways that work are <math>2,2,2,3,6</math> and <math>4,2,2,3,3</math>, for a total of <math>\tfrac{5!}{3!} + \tfrac{5!}{2!2!} = 20+30 = 50</math> possibilities. | ** For the case of no <math>1</math>, the two ways that work are <math>2,2,2,3,6</math> and <math>4,2,2,3,3</math>, for a total of <math>\tfrac{5!}{3!} + \tfrac{5!}{2!2!} = 20+30 = 50</math> possibilities. | ||
** For the case of one <math>1</math>, the three ways that work are <math>1,4,4,3,3</math> and <math>1,4,2,6,3</math> and <math>1,2,2,6,6</math>, for a total of <math>\tfrac{5!}{2!2!} + 5! + \tfrac{5!}{2!2!} = 30+120+30 = 180</math> possibilities. | ** For the case of one <math>1</math>, the three ways that work are <math>1,4,4,3,3</math> and <math>1,4,2,6,3</math> and <math>1,2,2,6,6</math>, for a total of <math>\tfrac{5!}{2!2!} + 5! + \tfrac{5!}{2!2!} = 30+120+30 = 180</math> possibilities. | ||
− | ** For the case of two <math>1</math>, the only way that works is <math>1,1,4, | + | ** For the case of two <math>1</math>, the only way that works is <math>1,1,4,6,6</math>, for a total of <math>\tfrac{5!}{2!2!} = 30</math> possibilities. |
Tallying up the results yields <math>240</math> ways to get <math>180</math> and <math>260</math> ways to get <math>144</math>, so the bigger probability is the product being <math>\boxed{144}</math>. | Tallying up the results yields <math>240</math> ways to get <math>180</math> and <math>260</math> ways to get <math>144</math>, so the bigger probability is the product being <math>\boxed{144}</math>. |
Latest revision as of 23:38, 30 January 2024
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.
- For the case of no
- 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.
- For the case of no
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 |