The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 153310+n, where n is a single digit natural number. Then, the value of (a+b+c+d+n) is
Question:
The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 153310+n, where n is a single digit natural number. Then, the value of (a+b+c+d+n) is
The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 153310+n, where n is a single digit natural number. Then, the value of (a+b+c+d+n) is
Options
Answer: 31
Explanation:
Step 1: Number of 4-digit numbers 4 distinct digits → 4! = 24 numbers Step 2: Contribution of each digit to total sum Each digit appears in each place (thousands, hundreds, tens, units) 6 times each, because 24 numbers ÷ 4 positions = 6. Total sum = 6 × (1000 + 100 + 10 + 1) × (a + b + c + d) = 6 × 1111 × (a + b + c + d) = 6666 × (a + b + c + d) Step 3: Match given sum We are told sum = 153310 + n So: 6666 × (a + b + c + d) = 153310 + n Divide 153310 by 6666 to get approximate (a+b+c+d): 6666 × 23 = 153318 Compare: 153310 + n = 153318 → n = 8 Then a + b + c + d = 23 Step 4: Compute a + b + c + d + n a + b + c + d + n = 23 + 8 = 31
Explanation:
Step 1: Number of 4-digit numbers 4 distinct digits → 4! = 24 numbers Step 2: Contribution of each digit to total sum Each digit appears in each place (thousands, hundreds, tens, units) 6 times each, because 24 numbers ÷ 4 positions = 6. Total sum = 6 × (1000 + 100 + 10 + 1) × (a + b + c + d) = 6 × 1111 × (a + b + c + d) = 6666 × (a + b + c + d) Step 3: Match given sum We are told sum = 153310 + n So: 6666 × (a + b + c + d) = 153310 + n Divide 153310 by 6666 to get approximate (a+b+c+d): 6666 × 23 = 153318 Compare: 153310 + n = 153318 → n = 8 Then a + b + c + d = 23 Step 4: Compute a + b + c + d + n a + b + c + d + n = 23 + 8 = 31
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