not sure if this is fair game on the GMAT but here's my answer:
the trick is isolating the repeating numbers (5 and 6), because these are the toughest to deal with. Based on this, there are 9 scenarios, but before we dive into them, please keep in mind a few things:
the word AND in counting problems is "multiplication" in math
when the word choose appears below, you're using the CHOOSE formula
when the word permute appears below, you're using the PERMUTATIONS formula
without further ado, the 9 scenarios are:
5 appears 0 times, 6 appears 0 times [0,0] --> choose 4 from the original 1,2,3,4 --> 4! = 24 ways to do this
5 appears 0 times, 6 appears 1 times [0,1] --> choose a spot for the "6", AND permute 3 from the original 1,2,3,4 --> 4 spots for "6" AND 24 permutations of 1,2,3,4 = 4*24 = 96
5 appears 0 times, 6 appears 2 times [0,2] --> choose 2 spots for the "6" AND permute 2 from the original 1,2,3,4 --> 6 spots for "6" AND 12 permutations of 1,2,3,4 = 6*12 = 72
5 appears 1 times, 6 appears 0 times [1,0] --> same math as [0,1] --> 96
5 appears 1 times, 6 appears 1 times [1,1] --> choose spots for the "6" AND "5", AND permute 2 from the original 1,2,3,4 --> 12 spots for "6" and "5" AND 12 permutations of 1,2,3,4 = 12*12 = 144
5 appears 1 times, 6 appears 2 times [1,2] --> choose 1 spot for "5" and 2 for "6" AND permute 1 from the original 1,2,3,4 --> 12 spots for "5" and "6" AND 4 permutations of 1,2,3,4 = 12*4=48
5 appears 2 times, 6 appears 0 times [2,0] --> same math as [0,2] --> 72
5 appears 2 times, 6 appears 1 times [2,1] --> same math as [2,1] --> 48
5 appears 3 times, 6 appears 2 times [2,2] --> choose 2 spots for the "6" and 2 for the "5"'s --> 6
now add those numbers together: 24+96+72 + 96+144+48 + 72+48+6 = 606
again, this is A LOT of work and not required, IMHO, in the GMAT, i'm going to go lie down now.
Dave
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