1994 AHSME Problems/Problem 30
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
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