SOLUTION: Find the sum of all positive six-digit integers whose digits are a permutation of the digits in the number 123456 without listing them all.

Question 1078654: Find the sum of all positive six-digit integers whose digits are a permutation of the digits in the number 123456 without listing them all.
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
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Answer. 111111*21*120.

Solution

In all, there are 6! = 1*2*3*4*5*6 = 720 such 6-digit numbers.

If you will write all these 720 6-digit numbers in one VERY long column, then

1. you will have 120 numbers that have "1" in the "ones" position;
120 numbers that have "2" in the "ones" position;
120 numbers that have "3" in the "ones" position;
120 numbers that have "4" in the "ones" position;
120 numbers that have "5" in the "ones" position;
120 numbers that have "6" in the "ones" position.

If you do sum all these 720 numbers, you will get the contribution to the sum from the "ones" position
of the value (1+2+3+4+5+6) = 21 repeated 120 times.

2. you will have 120 numbers that have "1" in the "tens" position;
120 numbers that have "2" in the "tens" position;
120 numbers that have "3" in the "tens" position;
120 numbers that have "4" in the "tens" position;
120 numbers that have "5" in the "tens" position;
120 numbers that have "6" in the "tens" position.

If you do sum all these 720 numbers, you will get the contribution to the sum from the "tens" position
of the value (1+2+3+4+5+6)*10 = 21*10 repeated 120 times.

3. you will have 120 numbers that have "1" in the "hundreds" position;
120 numbers that have "2" in the "hundreds" position;
120 numbers that have "3" in the "hundreds" position;
120 numbers that have "4" in the "hundreds" position;
120 numbers that have "5" in the "hundreds" position;
120 numbers that have "6" in the "hundreds" position.

If you do sum all these 720 numbers, you will get the contribution to the sum from the "hundreds" position
of the value (1+2+3+4+5+6)*100 = 21*100 repeated 120 times.

4. Similar for "thousands" position;

5. Similar for the "ten thousands" position;

6. Similar for the "hundred thousands" position.

Finally, you will have in the total sum
the contribution of 21*120 from the "ones" position;
the contribution of 21*120*10 from the "tens" position;
the contribution of 21*120*100 from the "hundreds" position;
the contribution of 21*120*1000 from the "thousands" position;
the contribution of 21*120*10000 from the "tens thousands" position;
the contribution of 21*120*100000 from the "hundred thousands" position.

Now sum 6 lines above, and you will get the final answer, which I wrote at the very beginning.

Solved.

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