# Problem of the Day: 1/10/13

The function lcm(a,b) gives the lowest common multiple of a and b. What is the value of lcm(lcm(8,12), lcm(9,12))?

Solution to yesterday’s problem:

Let’s use the blanks method as we did yesterday to find the number of possible arrangements first.

__8__ __7__ __6__ __5__ __4__ __3__ __2__ __1__

Multiplying these numbers together, we get 40,320 permutations (if not already, you should memorize these factorial numbers up to 10!). Now let’s cover all the cases in which the two E’s are consecutive. First, consider the case where the two E’s are at the beginning, and we’ll fill in the blanks after them.

__E__ __E__ __6__ __5__ __4__ __3__ __2__ __1__

There are 6! = 720 possible ways represented here. Now let’s move the E’s to the 2nd and 3rd positions. The number of ways represented will still be the same, because the numbers from above are just rearranged, not affecting their product. So we have 720 ways represented in each of these: 1st and 2nd blank, 2nd and 3rd blank, 3 and 4, 4 and 5, 5 and 6, 6 and 7, and 7 and 8. That’s 7 situations, and $720 \times 7=5040$.

So our number of occurrences with the E’s consecutive is 5,040, and the total number of occurrences is 40,320. Simplifying 5040/40320, we get the final probability 1/8.