2011 AMC 10A Problems/Problem 13
Problem 13
How many even integers are there between and whose digits are all different and come from the set ?
Solution
We split up into cases of the hundreds digits being or . If the hundred digits is , then the units digits must be in order for the number to be even and then there are remaining choices () for the tens digit, giving possibilities. Similarly, there are possibilities for the case, giving a total of possibilities.
Solution 2
We see that the last digit of the -digit number must be even to have an even number. Therefore, the last digit must either be or .
Case -the last digit is . We must have the hundreds digit to be and the tens digit to be any of , thus obtaining numbers total.
Case -the last digit is . We now can have or to be the hundreds digit, and any choice still gives us choices for the tens digit. Therefore, we have numbers.
Adding up our cases, we have numbers.
Solution 3 (elimination)
We see that there are total possibilities for a 3-digit number whose digits do not repeat and are comprised of digits only from the set Obviously, some of these (such as for example) will not work, and thus the answer will be less than The only possible option is ~ Technodoggo
See Also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
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Followed by Problem 14 | |
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