How many even four-digit integers smaller than 2000 are there such that all their digits are ...
How many even four-digit integers smaller than 2000 are there such that all their digits are different?
Answer/Solution
280
Steps/Work
Case 1: Numbers starting with 1 followed by an EVEN number
The hundreds digit can take 5 numbers (0,2,4,6,8) so 5 ways
The unit digit can take remaining 4 EVEN numbers left after using 1, to be EVEN so 4 ways.
The tens digit can take remaining 7 numbers left after using 3 numbers so 7 ways.
Total = 5*4*7=140
Case 2: Numbers starting with 1 followed by an ODD number
The hundreds digit can take 4 numbers (3,5,7,9) so 4 ways
The unit digit can take 5 numbers (0, 2, 4, 6, 8) to be EVEN, so 5 ways.
The tens digit can take remaining 7 numbers left after using 3 numbers so 7 ways.
Total = 4*5*7 = 140
Hence 140+140=280, correct answer is E
The hundreds digit can take 5 numbers (0,2,4,6,8) so 5 ways
The unit digit can take remaining 4 EVEN numbers left after using 1, to be EVEN so 4 ways.
The tens digit can take remaining 7 numbers left after using 3 numbers so 7 ways.
Total = 5*4*7=140
Case 2: Numbers starting with 1 followed by an ODD number
The hundreds digit can take 4 numbers (3,5,7,9) so 4 ways
The unit digit can take 5 numbers (0, 2, 4, 6, 8) to be EVEN, so 5 ways.
The tens digit can take remaining 7 numbers left after using 3 numbers so 7 ways.
Total = 4*5*7 = 140
Hence 140+140=280, correct answer is E