Difference between revisions of "1994 AHSME Problems/Problem 30"
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As such, minimizing <math>S</math> is the same as minimizing <math>n</math>. The minimum value of <math>n</math> is <math>\left\lceil \frac{1994}{6} \right\rceil = 333</math>. Hence, <math>S = 7n - 1994 = 337</math> <math>\textbf{(C)}</math>. | As such, minimizing <math>S</math> is the same as minimizing <math>n</math>. The minimum value of <math>n</math> is <math>\left\lceil \frac{1994}{6} \right\rceil = 333</math>. Hence, <math>S = 7n - 1994 = 337</math> <math>\textbf{(C)}</math>. | ||
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| + | ==See Also== | ||
| + | |||
| + | {{AHSME box|year=1994|num-b=29|after= Last Problem}} | ||
| + | {{MAA Notice}} | ||
Revision as of 17:28, 9 January 2021
Problem
When 
standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of 
. The smallest possible value of is
Solution
Given that there are dice, we know that the chance of rolling a sum of 
is the same as that of rolling a sum of 
. This is because there exists a bijection between the set 
of dice rolls that sum to and the set 
of dice rolls that sum to . In other words, for every ordered n-tuple 
such that 
for all valid 
and that 
, there is a unique ordered n-tuple 
where 
for all valid , whose sum is indeed .
As such, minimizing is the same as minimizing . The minimum value of is 
. Hence, 
.
See Also
| 1994 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 29 |
Followed by Last Problem | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
