Correct Answer:
Option A —
1/2
Here is the step-by-step breakdown using plain text without any mathematical symbols:
Step 1: Count the total valid outcomes (the given condition)
We are looking for outcomes where at least one 6 appears, but all three dice are not identical (which excludes the combination of three 6s).
Case 1: Exactly one 6 and two non-6s
The 6 can land on the first, second, or third die (3 possible positions).
The other two dice can be any number from 1 through 5 (5 choices each).
3 positions times 5 choices times 5 choices gives 75 outcomes.
Case 2: Exactly two 6s and one non-6
The non-6 can land on the first, second, or third die (3 possible positions).
The non-6 die can be any number from 1 through 5 (5 choices).
3 positions times 5 choices gives 15 outcomes.
Adding both cases together gives 90 total possible outcomes under the given condition.
Step 2: Count the outcomes where the sum is even
Since 6 is an even number, adding a 6 to a total does not change whether the total sum is even or odd.
For Case 1 (One 6 and two non-6s):
The sum will be even if the remaining two non-6 dice sum to an even number.
Two numbers sum to an even number if both are even or both are odd.
Both even (chosen from 2 or 4): 2 choices times 2 choices gives 4 ways.
Both odd (chosen from 1, 3, or 5): 3 choices times 3 choices gives 9 ways.
Total ways for the non-6 dice to sum to even: 4 plus 9 gives 13 ways.
Multiply by the 3 positions for the 6: 3 times 13 gives 39 favorable outcomes.
For Case 2 (Two 6s and one non-6):
Two 6s sum to twelve (an even number), so the remaining single non-6 die must be even.
The remaining die must be chosen from 2 or 4 (2 choices).
Multiply by the 3 positions for the non-6 die: 3 times 2 gives 6 favorable outcomes.
Adding the favorable outcomes from both cases gives 39 plus 6, which equals 45 favorable outcomes.
Step 3: Calculate the probability
Favorable outcomes: 45
Total given outcomes: 90
Probability: 45 out of 90, which simplifies to 1 out of 2 (one half).
Correct Answer:
A) 1/2
Answer verified by Quintessence Classes faculty — Karan Nagar, Srinagar.
