Expert: Paul Klarreich - 12/15/2008

Question
QUESTION: Hello,

With this question I was thinking that I should write out all of the possible solutions, but that seems very tedious, and I was wondering if there was a more mathematical way.

Thanks for any help you can offer.

Debbi

Suppose a state's license plates have 3 digits, then a picture of the state bird, and then 3 more digits.

(a) If any digit (0-9) can go in any position, how many different license plates are possible.

I was thinking maybe that if there are six places that can hold ten possibilities each, then it would be 60 multiplied by 6 because they can each be in any position.  so 60 x 6 = 360.

Please let me know if I am on the right track.

Thanks

ANSWER: Questioner:   debbi
Category:  Advanced Math
Private:  No
 
Subject:  ALGEBRA
Question:  Hello,

With this question I was thinking that I should write out all of the possible solutions, but that seems very tedious, and I was wondering if there was a more mathematical way.

Thanks for any help you can offer.

Debbi

Suppose a state's license plates have 3 digits, then a picture of the state bird, and then 3 more digits.

(a) If any digit (0-9) can go in any position, how many different license plates are possible.

I was thinking maybe that if there are six places that can hold ten possibilities each, then it would be 60 multiplied by 6 because they can each be in any position.  so 60 x 6 = 360.

Please let me know if I am on the right track.

Thanks
........................................
Hi, Debbi,

You must live in a very small state, or it must have a very large state bird.

Seriously,  you can have  10^6, or 1,000,000 plates.  All you are saying is that you can have any 6-digit number as a license plate.  Naturally, there are 10^6 of these, from 000000 to 999999.

Look up something called the Basic Counting Principle.


---------- FOLLOW-UP ----------

QUESTION: ok. There is a part B and for this question and I'm not sure how to do it.  Part B says if the state expects to use up all the license plate numbers within the next year,it needs to have a plan to develop more numbers.  If it decides to replace the first three digits with three letters form the alphabet  how many license plates are now possible with three letters and three digits? Repeating is allowed.  

Answer
Did you look up the counting principle?  It says:

If the first thing can be done n1 ways, and
If the second thing can be done n2 ways, and
If the third thing can be done n3 ways,

etc.

then the number of ways you can do them all is the PRODUCT of all those numbers.

So, you reason:

The first letter can be chosen  26 ways.
The next  letter can be chosen  26 ways.
The next  letter can be chosen  26 ways.
The first number can be chosen  10 ways.
The next  number can be chosen  10 ways.
The next  number can be chosen  10 ways.

What do you think comes next?