If a number is divisible by 4 and 5, then it is divisible by 20 (since 4 and 5 are relatively prime)….
A number is divisible by 20 if it ends in 20,40,60,80 or 00....
There exist two cases...
Numbers ending in 20,40,60,80, and numbers ending in 00…
I. Number of three-digit numbers divisible by 20 ending in 20,40,60,80 =4×9 =36...
Possible permutations of each of these three-digit number, including the number itself
=4 (e.g. 520→502,205,250) ...
Therefore, numbers in this case = 36×4 =144 ...
However, when both non-zero digits are even, the permutations are repeated.
(e.g. 820→802,208,280 and 280→802,208,820)
Such numbers =24×4=8 …
Subtract these cases. ⇒ total numbers in this case =144−8 =136 ..
- For those numbers ending in 00, no other permutations are possible.
Hence, numbers in this case =9 ..
Total cases
=136+9=145 ...
Total 3-digit numbers
=9×10×10=900 ....
Probability
=( 145/900 ) = ( 29 / 180 ) .…