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If you have one pair as in the example you sent 1,2&3,4,5 there are 5 choose 2 or 10 ways to choose a pair. They are
In each case there are 3 remaining buttons so you have a 4 stroke combination. The pair can be in any of the 4 positions and the remaining 3 buttons can be pushed in 3! = 6 possible orders. Thus if your combination has 1 pair there are
possible combinations. For example if 1&2 is the pair and it is in the first position the possible combinations are
If 1&2 is in the second position the combinations are
If there are 2 pairs in the combination, for example 1&2,3&4,5, then there are 5 choose 2 or 10 ways to choose the first pair and 3 choose 2 or 3 ways to choose a pair from the remaining 3 buttons. The one remaining button can be in 3 possible positions first, second or third. For example if you choose the pairs 1&2 and 3&4 then you have combinations
But there are twice this many possible combinations here since you can change the order of the pairs
Thus if the combination has 2 pairs then there are
possibilities. Hence if total there are 120 + 240 + 180 = 540 possible combinations. Good luck with this,  


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