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31 Mar 2024, 12:32
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D
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Difficulty:
95%
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Question Stats:
29% (02:00) correct
71%
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wrong
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For each set of three distinct nonzero digits, consider the sum of all positive three-digit integers that can be formed by the digits. For example, for the three digits 1, 2, and 3, the sum of all positive three-digit integers that can be formed by the digits is 123 + 132 + 213 + 231 + 312 + 321 = 1,332. How many different integers are equal to such a sum?
A. 9
B. 11
C. 14
D. 19
E. 24
A. 9
B. 11
C. 14
D. 19
E. 24
31 Mar 2024, 13:46
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I’m not able to solve it. Please help me understand the last line “ How many different integers are equal to such a sum?”
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Posted from my mobile device
31 Mar 2024, 14:17
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Bunuel
In each of the selected sets of three numbers, say \(x, y, \text{and}, z\) the value of the sum depends on the value of x, y, and z. This is because in each place we add two values of x + two values of y + two values of z + any carryover.
For example
123 + 132 + 213 + 231 + 312 + 321
To calculate the unit digit we must add (2 * 1) + (2 * 2) + (2 * 3) = 12
To calculate the tens digit we must add (2 * 1) + (2 * 2) + (2 * 3) + 1 (carry from units place) = 13
To calculate the hundreds digit we must add (2 * 1) + (2 * 2) + (2 * 3) + 1 (carry from tens place) = 13
Hence, we would obtain a different sum for different values of \(x + y + z.\)
Minimum value of \(x + y + z = 1 + 2 + 3 = 6\)
Maximum value of \(x + y + z = 7 + 8 + 9 = 24\)
Number of possible values = \(24 - 6 + 1 = 24 - 5 = 19\)
Clarification on the last part of the question "How many different integers are equal to such a sum?"
As mentioned above what matter is the possible values of \(x + y + z\). Different combinations of \(xyz\) can lead to the same sum if the value of \(x + y + z\) is the same. For example, \(xyz = 457\) and \(xyz = 367\) (or \(xyz = 187\)) will lead to the same sum as in all cases x + y + z = 16. We need to count the sum only once.
Option D
