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Problem 199

Combinatorics Difficulty 1.5 Multiple choice CEMC Gauss (Grade 7) · Canada · 2025

Two standard six-sided dice are rolled. If the two numbers on the
top faces are added, which of the following sums is least likely?

Pick one

Official solution

In the grid shown, the first column and first row show the
possible numbers that may appear on the top faces of the two dice. The
inside of the grid shows the sum of those corresponding two numbers.

$+$
$1$
$2$
$3$
$4$
$5$
$6$

$1$
$2$
$3$
$4$
$5$
$6$
$7$

$2$
$3$
$4$
$5$
$6$
$7$
$8$

$3$
$4$
$5$
$6$
$7$
$8$
$9$

$4$
$5$
$6$
$7$
$8$
$9$
$10$

$5$
$6$
$7$
$8$
$9$
$10$
$11$

$6$
$7$
$8$
$9$
$10$
$11$
$12$

Of the $36$ possible combinations
of rolls, a sum of $7$ can happen
$6$ different ways, a sum of $8$ can happen $5$ different ways, a sum of $9$ can happen $4$ different ways, a sum of $10$ can happen $3$ different ways, and a sum of $11$ can happen $2$ different ways. Thus, of the given
sums, $11$ is the least likely to
occur.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.