- Why are the numbers {1,7,8,9} not considered for a?
- Why are there only five positions when the numbers when 70000-20000=50000? I think this is because there are 5 even integers possible.
- Bonus (added after accepted answer): Why is bcd a permutation of P(3,(10-2)) if each one could contain {0,1,2,3,4,5,6,7,8,9}? Nevermind, it is because digits may not be repeated, and we lost two with a and e.
Let abcde be a required even integer.
Because {2,3,4,5,6} ∩ {0,2,4,6,8} = {2,4,6}, the problem can be divided into two cases.
Case 1: a ∊ {2,4,6}, therefore a has 3 choices, e has 4 (=5-1) choices 3 * 4 * P(3,8) = 4032
Case 2: a ∊ {3,5}, therefore a has 2 choices, e has 5 choices. 2 * 5 * P(3,8) = 3360
The total of even numbers is therefore 4032 + 3360 = 7392
From the book "Principles and Techniques in Combinatorics" by Chen Chuan-Chong and Koh Khee-Meng.
if a $5$-digit number begins with $1$, it is less than $20000$, if it begins with $7,8,9$, it is bigger or equal to $70000$.
All number between $20000$ and $70000$ consists of $5$ digits.