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09 Jun 2026, 03:28
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
32% (02:10) correct
68%
(02:36)
wrong
based on 25
sessions
History
Date
Time
Result
Not Attempted Yet
A circular spinner is divided into 9 equal sectors numbered 1 through 9. If the spinner is spun 3 times, how many different outcomes of the three spins result in a sum of 20?
A. 27
B. 36
C. 45
D. 72
E. 135
A. 27
B. 36
C. 45
D. 72
E. 135
09 Jun 2026, 19:53
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We need the number of ordered triples (a, b, c) such that:
a + b + c = 20
with each of a, b, c being an integer from 1 to 9.
Since the spins are separate, order matters.
Let
x = a − 1, y = b − 1, z = c − 1
Then x, y, z ≥ 0 and
x + y + z = 20 − 3 = 17
Also, since a, b, c ≤ 9,
x, y, z ≤ 8.
First count all nonnegative solutions to
x + y + z = 17.
Using stars and bars:
C(17 + 3 − 1, 3 − 1) = C(19, 2) = 171
Now subtract solutions where at least one variable exceeds 8.
Let x ≥ 9.
Set x' = x − 9 ≥ 0.
Then
x' + y + z = 8
Number of solutions:
C(10, 2) = 45
Similarly for y ≥ 9 and z ≥ 9, giving
3 × 45 = 135
Now add back intersections.
If x ≥ 9 and y ≥ 9, then
x' + y' + z = -1
impossible.
Likewise all pairwise and triple intersections are impossible.
Therefore:
Total valid solutions
= 171 − 135
= 36
Answer: Option: B 36
a + b + c = 20
with each of a, b, c being an integer from 1 to 9.
Since the spins are separate, order matters.
Let
x = a − 1, y = b − 1, z = c − 1
Then x, y, z ≥ 0 and
x + y + z = 20 − 3 = 17
Also, since a, b, c ≤ 9,
x, y, z ≤ 8.
First count all nonnegative solutions to
x + y + z = 17.
Using stars and bars:
C(17 + 3 − 1, 3 − 1) = C(19, 2) = 171
Now subtract solutions where at least one variable exceeds 8.
Let x ≥ 9.
Set x' = x − 9 ≥ 0.
Then
x' + y + z = 8
Number of solutions:
C(10, 2) = 45
Similarly for y ≥ 9 and z ≥ 9, giving
3 × 45 = 135
Now add back intersections.
If x ≥ 9 and y ≥ 9, then
x' + y' + z = -1
impossible.
Likewise all pairwise and triple intersections are impossible.
Therefore:
Total valid solutions
= 171 − 135
= 36
Answer: Option: B 36
