GATE DA 2026 Set 1 — Question 7
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Question
Five integers are picked from 0 to 20, with possible repetitions, such that their mean is 12, median is 18, and they have a single mode of 20.Ignoring permutations, the number of ways to pick these five integers is _____
Correct answer
(B) 1
Solution
Let the five integers be sorted in non-decreasing order: .
1.Median is 18: The middle element .
2.Single mode of 20: Since the list is sorted, the mode must appear as a block. For 20 to be the single mode, it must appear more frequently than any other number. Given , the values and must be 20. If only , the frequency is 1, which cannot be a unique mode if others are distinct, or ties with others. If , the frequency is 2. Since , 20 cannot appear 3 times. Thus, . The set is .
3.Mean is 12: The sum of the integers is .
Possible pairs such that :- Case 1: . Set: . Frequencies: . Mode is 20 (unique). Valid.
- Case 2: . Set: . Frequencies: . Modes are 1 and 20. Invalid (must have a single mode).
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