How many even numbers less than 500 can be formed using the digits 1, 2, 3, 4 and 5? Each digit may be used only once in any number.
Question
How many even numbers less than 500 can be formed using the digits 1, 2, 3, 4 and 5? Each digit may be used only once in any number.
Solution
To solve this problem, we need to consider the different cases for the number of digits in the number.
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One-digit numbers: The only even one-digit number is 2. So, there is 1 such number.
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Two-digit numbers: The units place must be an even number. So, there are 2 choices (2 and 4). The tens place can be any of the remaining 4 digits. So, there are 4 choices. Therefore, there are 2*4 = 8 such numbers.
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Three-digit numbers: The units place must be an even number. So, there are 2 choices. The tens place can be any of the remaining 4 digits. So, there are 4 choices. The hundreds place can be any of the remaining 3 digits. So, there are 3 choices. Therefore, there are 243 = 24 such numbers.
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Four-digit numbers: The units place must be an even number. So, there are 2 choices. The tens place can be any of the remaining 4 digits. So, there are 4 choices. The hundreds place can be any of the remaining 3 digits. So, there are 3 choices. The thousands place can be any of the remaining 2 digits. So, there are 2 choices. Therefore, there are 243*2 = 48 such numbers.
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Five-digit numbers: The units place must be an even number. So, there are 2 choices. The tens place can be any of the remaining 4 digits. So, there are 4 choices. The hundreds place can be any of the remaining 3 digits. So, there are 3 choices. The thousands place can be any of the remaining 2 digits. So, there are 2 choices. The ten-thousands place can be any of the remaining 1 digit. So, there is 1 choice. Therefore, there are 24321 = 48 such numbers.
Adding up all these cases, there are 1 + 8 + 24 + 48 + 48 = 129 even numbers less than 500 that can be formed using the digits 1, 2, 3, 4, and 5 each only once.
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