120 of the possible 720 numbers are divisible by 5.
We are to form 6-digit numbers from the digits 1,2,3,4,5, and 6 using each number exactly once. The total number of such numbers is 720. We can get this using the fundamental counting principle; there are 6 choices for the first digit, then 5 choices for the second digit, then 4 choices for the third digit, 3 choices for the fourth digit, 2 choices for the fifth digit, and 1 choice for the last digit. Then 6x5x4x3x2x1=720.
Alternatively if we select 6 items from 6 items without repetition and ask for the number of possibilities, we see that order matters, so using permutations we find that `_6P_6=720` . Also we could know that the number of arrangements of n objects, selecting all of them, is n! so 6!=720.
We are asked how many of these numbers are divisible by 5. A number is divisible by 5 if and only if the last digit is 0 or 5. Since none of the numbers end in 0, we need only determine how many end in 5.
There are 5 possibilities for the first five digits of the number (5 is the last digit), so there are 5!=120 different such numbers.