Count the number of the even integers in the range of $2000$ to $7000$ such that each number doesn't have repeated digits.

Question

Count the number of the even integers in the range of $2000$ to $7000$ such that each number doesn't have repeated digits.

I count the numbers with computer, the result was $1232$.

Answer 1

For a number to be even, its units digit must be $0$, $2$, $4$, $6$, or $8$.

Since $7000$ has a repeated $0$, the thousands digit must be $2$, $3$, $4$, $5$, or $6$.

With this in mind, we consider cases, depending on whether or not the units digit is a potential thousands digit.

Case 1: The units digit is $0$ or $8$

We have two choices for the units digit. Since the units digit is not among the five choices for the thousands digit, we still have five choices for the thousands digit. This leaves us with eight choices for the hundreds digit (since the units digit and thousands digit are excluded) and seven choices for the tens digit (since the units digit, thousands digit, and hundreds digit are excluded). There are $$5 \cdot 8 \cdot 7 \cdot 2 = 560$$ such numbers.

Case 2: The units digit is $2$, $4$, or $6$.

We have three choices for the units digit. Whichever one we choose eliminates one of the five choices for the thousands digit, leaving us with four choices for the thousands digit. This leaves with eight choices for the hundreds digit and seven choices for the tens digit. There are $$4 \cdot 8 \cdot 7 \cdot 3 = 672$$ such numbers.

Total: Since the two cases are mutually exclusive, the number of even numbers between $2000$ and $7000$ with no repeated digits is $$560 + 672 = 1232$$ as you found with a computer search.

Answer 2

How many even integers are there with 2 repeated digits? How many even integers are there with 3 repeated digits? How many even integers are there with 4 repeated digits?