2023 AMC 12B Problems/Problem 23
Problem
When standard six-sided dice are rolled, the product of the numbers rolled can be any of
possible values. What is
?
Solution1
The product can be written as
Therefore, we need to find the number of ordered tuples where
,
,
,
,
are non-negative integers satisfying
.
We denote this number as
.
Denote by the number of ordered tuples
where
with
.
Thus,
Next, we compute .
Denote . Thus, for each given
, the range of
is from 0 to
.
Thus, the number of
is
Therefore,
By solving , we get
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2 (Cheese)
The product can be written as
Letting , we get
, 6 possible values. But if the only restriction of the product if that
, we can get
possible values. We calculate the ratio
Letting , we get
, 17 possible values.
The number of possibilities in the ideal situation is
, making
.
Now we can predict the trend of : as
increases,
decreases.
Letting
, you get possible values of ideal situation=
.
, the number=
.
, the number=
.
, the number=
so 6 is not the answer.
, the number=
.
, the number=
,but
≈
still much smaller than 936.
, the number=
,but
≈
still smaller than 936.
, the number=
,
≈
is a little bigger 936, but the quotient of (possible values of real situation)/(possible values of ideal situation) is much smaller than 0.378 now, so 10 is probably not the answer,so the answer is
.
Check calculation:
,the number=
,
≈
is much bigger than 936.
~Troublemaker
Video Solution 1 by OmegaLearn
See Also
2023 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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