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Permute It

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

To add up all the permutations of all the 5 digit numbers written using the digits 1, 2, 3, 4 and 5 just once each time. A good way to tackle problems is to try simple cases and here you might try 21 + 12 = 33 then try it with 3 digits and so on. Ong Xing used the following method.

There are 5! numbers, and in each column of the sum each of the digits 1, 2, 3, 4 and 5 appears 4! = 24 times. The sum of each column is therefore 24(1 + 2 +3 + 4 +5) = 24 x 15 = 360.

The total sum is 360(1 + 10 + 102 + 103 + 104) = 360 x 11111 = 399960.

Joel used this even shorter argument. The number of permutations is 5!, that is 120. The average is 3, so the total is 33333 x 120 = 3999960.