State License Plates

Question

In a certain state license plate consists of three digits followed by 4 letters.

 

Find:

a. The Number of license plates possible if repetition is allowed.

b. The number of license plates possible if the last letter must be an M and the last digit must be a 5 repetition is not allowed

 

 

P.S. A and B are different problems that need to be answer with the same situation with three digits and 4 letters.

Answer 1

Ethan,

There are 10 choices for each of the 3 single digit positions: 0, 1, 2, ..., 9.

There are 26 choices in English for each of the 4 single letter positions if only caps are permitted: A, B, C, ..., Z. That would expand to 52 if lower case letters were also allowed, but let's assume not.

 

(a) If repetition is allowed in each of the 7 positions of a license plate, then the 

total # of plates possible = (10)³ (26)⁴ since the choices for each type of position are identical and you just multiply them all together.

 

(b) No repetition is permitted and plates must satisfy pattern:  xx5xxxM.  Imagine you have a set of wooden blocks with each block being a distinct number or a cap letter. Then once you use the block in a plate position, it is no longer available for any other position.

 

Now you start with only 9 digit choices for position1 since 5 has already been used in position3. And then only 8 choices for position2.

Thus #numerical patterns = 9 (8) (1) = 72.

 

Similarly in the lettered area of the plate, the choices get reduced each time a position gets filled:

Thus # lettered patterns = (25)(24)(23)(1) = 13,800

 

Finally, the total # plate combinations = 72 (13,800) = 993,600.

Answer 2

This is a simple permutations problem. A simple way to picture this is to set up places for each letter/number like this:

 

_ _ _ _ _ _ _

 

And then putting in the appropriate numbers. For the first place, the character must be a digit, and there are 10 digits from which we can choose (0-9), so 10 goes in the first place.

 

10 _ _ _ _ _ _

 

For the second place, since repetition is allowed, we also have 10 digits from which we can choose for it as well, and the same is true for the third place, so 10 is inputted into those places:

 

10 10 10 _ _ _ _

 

For the fourth place, we have 26 letters from which we can choose, so 26 is put in that place. Since repetition is allowed, we also have 26 letters for the fifth, sixth, and seventh places as well, so:

 

10 10 10 26 26 26 26

 

And then we multiply those numbers together to get 456,976,000, and that is the number of possible permutations of license plates with the above set of criteria.

 

For the second part, we do the same thing:

 

_ _ _ _ _ _ _

 

If we start with the numbers, we only have 9 to choose from for the first place, since the number 5 is set to be used for the last numbers place and repetition is not allowed, so:

 

9 _ _ _ _ _ _

 

For the second place, we only have 8 from which we can choose, since repetition is not allowed. For the third, we only have one from which we can choose, since the last digit MUST be a digit that has already been chosen, so:

 

9 8 1 _ _ _ _

 

For the letters, we have 25 from which we can choose, since the last letter has already been chosen, and repetition is not allowed:

 

9 8 1 25 _ _ _

 

For the fifth place, we have 24 from which we can choose, and for the sixth, we have 23. For the seventh, we have only one, since it has already been chosen, so:

 

9 8 1 25 24 23 1

 

Now we multiply those numbers together, and we get 993600 possible permutations.

Answer 3

Hello Ethan,

 

This isn't as difficult as it seems. It's really just multiplication.

 

For A, we know that there are 3 digits than 4 letters and they can be any digits or letters. We know the alphabet has 26 letters and the digits can be from 0 to 9 which is 10 total digits. With that said we do the following:

 

10 * 10 * 10 * 26 * 26 * 26 * 26 = 456,976,000

 

For B, there is no repetition so we don't multiply the same way.  In fact, we are asked about a certain event which we do independently of each other so we do the digits first by multiplying by one less from the last term so as to take out the idea of a repetition. We need to multiply with (x-1), meaning the total number of digits or alphabets minus one. Here is what we do:

 

9 * 8 * 25 * 24 * 23 = 993,600

 

We do this because we know we need M so we know that there can only be 25 other letters than since there is no repetition allowed. From that, there are three letters and they can't repeat so we go from 25 *(25-1) * (24-1) and for the digits, it's the same concept. We know 5 is the last digit so there are only 9 digit possibilities left so we do 9 * 8 and multiply everything together. 

 

I hope this helps. :)

 

 

 

 

 

Answer 4

Hi Ethan,

 

a)

 

to get the first part we have ten possible choices and then we pick one and we have nine choices and then we pick one and we have 8 choices

 

so 10*9*8

 

to get the second part we have  26 choices (letters) then 25 then 24 then 23

 

so 26*25*24*23

 

 

OK now multiply  all of these

 

and we get a lot of license plates  258,336,000

 

 

b)

 

this is similar but a couple choices are restricted

 

so for our digits we have 9 * 8 *1  (because 5 must be the last digit - it can't be the 1st or second)

 

and for our letters

 

we have 25 * 24 * 23 * 1 (m must be our last letter and it can't be 1st 2nd or 3rd)

 

and when we multiply these we get 993,600

 

hope this helps