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For each set of three distinct nonzero digits, consider the sum of all
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Updated on: 15 Mar 2023, 12:01
Updated on: 15 Mar 2023, 12:01
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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
Originally posted by Kimberly99 on 15 Mar 2023, 11:57.
Last edited by Kimberly99 on 15 Mar 2023, 12:01, edited 1 time in total.
Last edited by Kimberly99 on 15 Mar 2023, 12:01, edited 1 time in total.
Re: For each set of three distinct nonzero digits, consider the sum of all
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15 Mar 2023, 12:00
15 Mar 2023, 12:00
Hi Carcass, could you help with this question please? Thanks a bunch.
Re: For each set of three distinct nonzero digits, consider the sum of all
[#permalink]
15 Mar 2023, 12:18
15 Mar 2023, 12:18
Expert Reply
The sum of the 6 different permutations of abc =
100a+10b+c
+100a+10c+b
+100b+10a+c
+100b+10c+a
+100c+10a+b
+100c+10b+c
=222(a+b+c)
Therefore, the sum of a+b+c determines the sum of the 6 different permutations of abc
How many different sums of a+b+c are there, and how many sums of 6 different permutations of abc are there?
The minimum case of the sum of a+b+c, 1+2+3=6, the sum of six different permutations=6*222
The maximum case of the sum of a+b+c, 7+8+9=24, the sum of six different permutations=24*222
1, 2, 3, 4, 5, 6, 7, 8, 9 numbers are used arbitrarily, then the sum of a+b+c can be obtained continuously from 6 to 24, and there are 24-6+1=19 possibilities
There are 19 different sums of a+b+c, so there are 19 sums of 6 different permutations of abc
100a+10b+c
+100a+10c+b
+100b+10a+c
+100b+10c+a
+100c+10a+b
+100c+10b+c
=222(a+b+c)
Therefore, the sum of a+b+c determines the sum of the 6 different permutations of abc
How many different sums of a+b+c are there, and how many sums of 6 different permutations of abc are there?
The minimum case of the sum of a+b+c, 1+2+3=6, the sum of six different permutations=6*222
The maximum case of the sum of a+b+c, 7+8+9=24, the sum of six different permutations=24*222
1, 2, 3, 4, 5, 6, 7, 8, 9 numbers are used arbitrarily, then the sum of a+b+c can be obtained continuously from 6 to 24, and there are 24-6+1=19 possibilities
There are 19 different sums of a+b+c, so there are 19 sums of 6 different permutations of abc
