Question

Find the sum of all the numbers that can be formed with the digits 2,3,4,5 taken all at a time.

Answer 1

By symmetry, each digit will occur equally often in each position. Thus, the average number in each digit position will be (2+3+4+5)/4 = 14/4 = 7/2. There are 24 permutations. So we get a sum of (24*7/2) = 84 in each position, which means the total is 84+840+8400+84000 = 93324. 

Another way of looking at it is that for each digit in each spot, if we fix that digit, there are 3! ways to arrange the four numbers with that digit fixed. So in the sum you get 2*3! + 3*3! + 4*3! + 5*3! = (2+3+4+5)*3! = 14*3! = 14*6 = 84. So again, it's 84000+8400+840+84 = 93324.


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Answer 2

2+3+4+5=14.
2 + 3 = 5
2 + 4 = 6
2 + 5 = 7
3 + 4 = 7
4 + 5 = 9
5 + 3 = 8