Question

How many even integers between $4000$ and $7000$ have four different digits?

Answer

Answer: 728

Concepts:basic counting casework digits

Difficulty rating: 1920

Small Hint:

Separate the cases according to whether the thousands digit is even or odd

Big Hint:

After choosing the thousands and units digits, count the choices for the two middle digits in order

Solution:

The thousands digit is $4,$ $5,$ or $6.$ If it is $4$ or $6,$ the units digit has $4$ choices among $0,$ $2,$ $4,$ $6,$ and $8,$ after which the hundreds and tens digits have $8$ and $7$ choices. These two cases contribute $2\cdot4\cdot8\cdot7=448.$ If the thousands digit is $5,$ all $5$ even units digits are available, contributing $5\cdot8\cdot7=280.$ Thus the total is $448+280=728.$

1993 AIME Problem 1

Answer: 728

Difficulty rating: 1920

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years