Hello, lawochekel!

I'll explain it using baby-steps.

Five-digit numbers are formed by using the digits {0,1,2,3,4,5} without repetition.

Find the probability that the number is divisible by 6.

There are possible five-digit numbers.

To be divisible by 6, the number must be:

. . (1) divisible by 3 ... the sum of its digits must be divisible by 3.

. . (2) even ... the last digit must be even.

There are two sets of digits whose sums are divisible by 3:

. .

Consider

The last digit must be even: , 3 choices.

The other four digits can be arranged in ways.

Hence, there are:. numbers divisible by 6.

Consider

The last digit must be even: , 2 choices.

The other four digits can be arranged in ways.

Hence, there are:. more numbers divisible by 6.

Hence, there are:. numbers which are divisible by 6.

Therefore:.