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2011 AMC 10A Problem 13

Problem 13 of 25IntermediateNumber TheoryCombinatoricsProblem-Solving Techniques

How many even integers are there between 200200 and 700700 whose digits are all different and come from the set {1,2,5,7,8,9}?\{1,2,5,7,8,9\}?

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Solution

Since the hundreds digit can only be a 22 or 5,5, we can case on this value. Case 1:1: hundreds digit is 22 The only option for the units digit is 8,8, since the number must be even. This leaves 44 options for the tens digit. This gives us 1⋅4=41 \cdot 4 = 4 numbers for this case. Case 2:2: hundreds digit is 55 Similarly to above, 22 and 88 are the only options for the units digit, leaving 44 options for the tens digit. This gives us 2⋅4=82 \cdot 4 = 8 numbers for this case. The total number of integers is then 4+8=12.4 + 8 = 12. Thus, A is the correct answer.
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Tagged: digits · casework · multiplication principle

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