How many license plates can be produced using 5 symbols on each plate where the first 2 symbols are letters of the alphabet and the following three symbols

Question

How many license plates can be produced using five symbols on each plate where the first two symbols are letters of the alphabet and the following three symbols are numbers selected from the set ?{0,1,2,3,4,5,6,7,8,9}? No number or letter can be repeated on the same license plate.

Answer 1

Hello Amanda,

 

Both answers provided by Roman C. and Mark M. are correct, so I am just adding couple words in regards to computational aspects. Known from the basic combinatorics, the formula for all possible permutations k (where order matters) out of n can be written as:

 

P(n,k) = n! / (n-k)!

or

P(n,k) = n(n-1)...(n-k+1)

 

Both formulas theoretically provide identical results, however in many practical cases the calculation of n! on personal computer may cause an overflow error. Thus, the second computational method (it essentially calculates a product of numbers in decreasing arithmetic sequences) is preferable in most cases due to its computational efficiency.

 

Best regards,

Answer 2

Recall that the number of ways of choosing k objects from a set of distinct objects when order matters is

 

P(n,k) = n! / (n-k)! = n(n-1)...(n-k+1)

 

Why?

 

There are n ways to choose the first object, then n-1 ways to choose the second, and so on.

 

There are P(26,2) = 26·25 = 650 ways to choose the string of letters.

 

There are P(10,3) = 10·9·8 = 720 ways to choose the string of numbers.

 

Since these two parts are independent and the numbers are concatenated after the letters the number of license plates possible is

 

650·720 = 468000

Answer 3

There are 26 letters in the alphabet, so there are 26 choices for the first letter.  Letters can't be repeated, so only 25 choices are available for the second letter.  There are 10 choices for the first number.  Since numbers can't be repeated, there are 9 choices for the second number and 8 choices for the third.  So, the number of possible license plates is:

 

(26)(25)(10)(9)(8) = 468,000