GATE DA 2024 Set 1 — Question 3
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General Aptitude → Counting → Combinations
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Question
How many 4-digit positive integers divisible by 3 can be formed using only the digits , such that no digit appears more than once in a number?
Correct answer
(B) 48
Solution
We need to form 4-digit numbers using distinct digits from the set that are divisible by 3.Rule for divisibility by 3: The sum of the digits of the number must be divisible by 3.
Sum: (Divisible by 3)
Number of permutations:
Sum: (Divisible by 3)
Number of permutations:
1.Calculate the sum of all available digits:
Since 21 is divisible by 3, to form a 4-digit subset with a sum divisible by 3, the single digit we exclude must also be divisible by 3 (because is a multiple of 3 only if is a multiple of 3).2.Identify digits divisible by 3 in the set :
The candidates are 3 and 6.3.Case 1: Exclude 3
Digits used: Sum: (Divisible by 3)
Number of permutations:
4.Case 2: Exclude 6
Digits used: Sum: (Divisible by 3)
Number of permutations:
5.Total numbers:
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