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Directions: Find the sum of all 4 digit numbers formed by using the digits 2, 3, 6, 9 in which no digit is repeated.
- a)122320
- b)133480
- c)133230
- d)133320
- e)None of the Above
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Directions: Find the sum of all 4 digit numbers formed by using the d...
Consider 2 in the units digit: The other digits can be arranged in 3! ways. Similarly, if 2 is in tens, hundreds, thousands place, we get 3! numbers.
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Contribution of 2 to the overall sum = 2 × 3! × (1 + 10 + 100 + 1000) = 2 x× 3! × 1111
Similarly contribution of 3, 6, 9 will be 3 × 3! × 1111, 6 × 3! × 1111 and 9 × 3! × 1111 respectively.
Overall sum = (2 + 3 + 6 + 9) × 3! × 1111 = 133320
Most Upvoted Answer
Directions: Find the sum of all 4 digit numbers formed by using the d...
Understanding the Problem
To find the sum of all 4-digit numbers formed using the digits 2, 3, 6, and 9 without repetition, we first determine how many unique numbers can be formed and then calculate their contributions to the overall sum.
Number of Combinations
- The total number of 4-digit permutations using the digits 2, 3, 6, and 9 is 4! (factorial of 4).
- This equals 24 unique numbers.
Position Contribution
Each digit will appear in each position (thousands, hundreds, tens, units) equally across all permutations.
- Each digit appears in each position 6 times (24 total numbers divided by 4 positions).
Calculating Contribution
1. For Each Position:
- Thousands: Each digit contributes its value times 1000.
- Hundreds: Each digit contributes its value times 100.
- Tens: Each digit contributes its value times 10.
- Units: Each digit contributes its value times 1.
2. Sum of the Digits:
- The sum of the digits 2, 3, 6, and 9 = 2 + 3 + 6 + 9 = 20
Total Contribution Calculation
- Contribution per position:
- Thousands: 6 * (20 * 1000) = 120000
- Hundreds: 6 * (20 * 100) = 12000
- Tens: 6 * (20 * 10) = 1200
- Units: 6 * (20 * 1) = 120
Total Sum
- Adding these contributions:
- Total = 120000 + 12000 + 1200 + 120 = 133320
Thus, the correct answer is option D: 133320.
To find the sum of all 4-digit numbers formed using the digits 2, 3, 6, and 9 without repetition, we first determine how many unique numbers can be formed and then calculate their contributions to the overall sum.
Number of Combinations
- The total number of 4-digit permutations using the digits 2, 3, 6, and 9 is 4! (factorial of 4).
- This equals 24 unique numbers.
Position Contribution
Each digit will appear in each position (thousands, hundreds, tens, units) equally across all permutations.
- Each digit appears in each position 6 times (24 total numbers divided by 4 positions).
Calculating Contribution
1. For Each Position:
- Thousands: Each digit contributes its value times 1000.
- Hundreds: Each digit contributes its value times 100.
- Tens: Each digit contributes its value times 10.
- Units: Each digit contributes its value times 1.
2. Sum of the Digits:
- The sum of the digits 2, 3, 6, and 9 = 2 + 3 + 6 + 9 = 20
Total Contribution Calculation
- Contribution per position:
- Thousands: 6 * (20 * 1000) = 120000
- Hundreds: 6 * (20 * 100) = 12000
- Tens: 6 * (20 * 10) = 1200
- Units: 6 * (20 * 1) = 120
Total Sum
- Adding these contributions:
- Total = 120000 + 12000 + 1200 + 120 = 133320
Thus, the correct answer is option D: 133320.
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