Divisibility Rule Check
A number is divisible by 3 if the sum of its digits is divisible by 3.
The available digits are {1, 3, 4, 6, 7}. The sum of these digits is $1 + 3 + 4 + 6 + 7 = 21$.
We need to form 4-digit integers using these digits without repetition. This requires selecting 4 digits from the given 5.
Let the sum of the 4 chosen digits be $S_4$. The sum of all 5 digits is $S_5 = 21$. If we exclude one digit $d$, the sum of the remaining 4 digits is $S_4 = S_5 - d = 21 - d$.
For $S_4$ to be divisible by 3, $21 - d$ must be divisible by 3. Since 21 is divisible by 3, the excluded digit $d$ must also be divisible by 3.
Identifying Usable Digit Sets
From the set {1, 3, 4, 6, 7}, the digits divisible by 3 are 3 and 6.
- Case 1: Exclude digit 3. The remaining digits are {1, 4, 6, 7}. Their sum is $1 + 4 + 6 + 7 = 18$, which is divisible by 3.
- Case 2: Exclude digit 6. The remaining digits are {1, 3, 4, 7}. Their sum is $1 + 3 + 4 + 7 = 15$, which is divisible by 3.
Counting Permutations
For each set of 4 digits, we find the number of distinct 4-digit integers that can be formed. Since the order of digits matters and repetition is not allowed, this is a permutation calculation.
- Case 1 (Digits {1, 4, 6, 7}): The number of possible 4-digit integers is $4!$. $4! = 4 \times 3 \times 2 \times 1 = 24$.
- Case 2 (Digits {1, 3, 4, 7}): The number of possible 4-digit integers is $4!$. $4! = 4 \times 3 \times 2 \times 1 = 24$.
Total Count Calculation
The total number of 4-digit positive integers divisible by 3 is the sum of the counts from the valid cases.
Total Integers = (Count from Case 1) + (Count from Case 2)
Total Integers = $24 + 24 = 48$.
Important Questions from Permutations and Combinations
Three distinct sets of indistinguishable twins are to be seated at a circular table that has 8 identical chairs. Unique seating arrangements are defined by the relative positions of the people.
How many unique seating arrangements are possible such that each person is sitting next to their twin?
- Mixed species flocks of birds include social and solitary species. There are 5 social species and 10 solitary species in a forest. Flocks always have a total of 5 species, of which 2 are social and 3 are solitary. The number of types of flocks with unique species composition is ________.(Answer in integer)
- How many five-digit numbers can be formed using the integers 3, 4, 5 and 6 with exactly one digit appearing twice?
Two wizards try to create a spell using all the four elements, water, air, fire, and earth. For this, they decide to mix all these elements in all possible orders. They also decide to work independently. After trying all possible combination of elements, they conclude that the spell does not work.
How many attempts does each wizard make before coming to this conclusion, independently?Four persons P, Q, R and S are to be seated in a row. R should not be seated at the second position from the left end of the row. The number of distinct seating arrangements possible is:
