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Another way to approach this problem is to recognize that the way the sequence increases from the min (1234) is symmetrical to the way it decreases from the max (5432). Therefore if you find the average of the min and max and multiply it by the number of possibilities (5*4*3*2 or 120) then you'll have your answer.

\(\frac{1234+5432}{2} = 3333\)

\(3333*120 = 399960\)

(2)

You can use the same method here. The min is 1111 and the max is 5555. The number of possibilities is 5*5*5*5 = 625

\(\frac{1111+5555}{2} = 3333\)

\(3333*625 = 2083125\)

(3)

For this one we need to be a little more clear. Would the sum include duplicate numbers or should all numbers be distinct? For example, the number 1223 can be formed two ways. The first way would be 1 (1st 2) (2nd 2) 3. The other way would be 1 (2nd 2) (1st 2) 3. If all numbers need to be distinct then the solution will be difficult. I used excel to find the sum to be 933240. I would be very interested if someone knows how to do this problem assuming all numbers must be distinct. Anyways, the method I used above won't work since you lose symmetry. Instead you can evaluate using Bunuel's solution like so:

There are seven numbers that we can put into a group of four: 1, 2, 2, 3, 4, 5, 5 The number of possibilities is 7*6*5*4 = 840 840/7 = 120 Therefore the sum = \(1111*(120*(1+2*2+3+4+2*5)) = 2933040\)

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