When I’m stuck in traffic, I have random thoughts racing through my mind since my stereo/radio is out of commission. After all, what else is there to do when your car is moving at such a slow pace?
This morning, while I was crawling along with the throng of cars, I began wondering just how many possible license plate number combinations exist as I glanced from one vehicle to another.
In California, there are seven spots available for the license plate number. The usual combination is a number, three letters and three more numbers. So for example, a typical license plate would read 1AAA111.
To satisfy my curiosity, I decided to calculate the permutation.
Since there are ten available numbers–zero through nine–and twenty-six letters in the alphabet, that gives me a total of thirty-six characters. I do not account for personalized plates that have fewer than seven numbers/letters or state vehicles that have plates fewer than seven numbers/letters. I calculated the permutation as follows: 36!/29!.
So what is the final number? It turns out there are 42,072,307,200 different possible license plate combinations for California automobiles. I sure hope we never see forty billion cars in California because that would make traffic even worse!
Update: My friend, Sam, pointed out to me that this problem should not be solved with a permutation. Instead, it can be solved by simple multiplication. He left notes below regarding how it can be solved correctly based on the DMV criteria. However, I have seen trucks where the license plate’s fourth spot is occupied by a number. Given this criteria and making no exceptions when it comes to letters (i.e., I will allow the letters I, O and Q to be used in any of the spots where numbers usually occupy. Also, I will allow numbers to be used in the fourth spot. That brings the number of combinations to 10*26*26*36*10*10*10, which equals 243,360,000. However, if we include the DMV’s restriction and allow for the numbers to occupy the 4th spot, we have 10*23*26*33*10*10*10, which equals 197,340,000 combinations.