License Plate Permutation Problem

[Chris_Green]

Problem: A Canadian postal code consists of 6 characters of 3 letters alternating with 3 digits. An example is M4N 0R3. Mo thought it would be fun to have a postal code that has his name somewhere in it (the letters M and O in that order) and at least one 7, his favourite number. Determine the probability of this occurring.

*I'm having trouble finding the number of possibilities that are possible under these circumstances but I know for sure that the total number of possibilities are 26^3 x 10^3.
Help would be much appreciated

 

[Dr.Peterson]

Think about how you might go about making such a code.

Pick a pair of places for the M and O. How many ways can that be done?

Pick a letter for the other letter slot. How many ways?

Now think about how many ways there are to fill the three digit slots with at least one being a 7. (You might want to think about the number of ways to do it without a 7 first!)

Let us know what you've done, so we can commend or correct you.

 

[Chris_Green]

The numbers I understand. The total possibilities without a 7 is simply 9^3 making the the possibilities with a 7: 10^3 - 9^3 but I'm really confused on how to get the possibilities of letters. I know the possible arrangements are M*O***, **M*O* or M***O*.

 

[Steven G]

The numbers I understand. The total possibilities without a 7 is simply 9^3 making the the possibilities with a 7: 10^3 - 9^3 but I'm really confused on how to get the possibilities of letters. I know the possible arrangements are M*O***, **M*O* or M***O*.

Do you think that putting all 3 letters first and then numbers next yield the same answer as your problem.

Consider the letters fill in _ _ _ where MO has to be together. How many ways can this be done?

Do a similar thing with the 3 numbers

 

[Dr.Peterson]

The numbers I understand. The total possibilities without a 7 is simply 9^3 making the the possibilities with a 7: 10^3 - 9^3 but I'm really confused on how to get the possibilities of letters. I know the possible arrangements are M*O***, **M*O* or M***O*.

You've practically got the answer!

You've listed the only three ways the letters can be placed (in order, but not necessarily consecutive); in Jomo's terms (which is exactly how I think of it), the three blanks for letters can be filled as MO_, M_O, or _MO, and nothing else. So there are 3 ways to place the letters.

Now combine that with the ways to place the digits, which you have correct.

 

[Steven G]

I wonder if Mo would like the code to be m4q6o7, but I guess that the problem allows for this.

 

[Dr.Peterson]

It seems to, though one could argue otherwise. What I was wondering about when I first read it was something like M0A7B1!