Suppose a license plate consists of 3 letters and 3 digits. How many license plates can be made?

Question

Is the answer 26*26*26*10*10*10 to signify that there must be exactly 3 letters and exactly 3 numbers? Would this equation work similarly if the order of the letters and numbers did not matter (ex: Letter, Number, Number, Letter, Letter, Number)? The supposed answer is 36*36*36*36*36*36, or 36^6. Is there any proof behind this?

Answer 1

Choose the 3 positions where you put the letters:

 

C(6,3) = 6!/(3!3!) = 720/36 = 20

 

Then choose any letter for each of those positions:

 

26³ = 17,576

 

Then choose any digit for each of the other positions:

 

10³ = 1,000

 

Total number of license plates:

 

20 × 17,576 × 1,000 = 351,520,000

Answer 2

Note a proof, yet some reasoning.

Your first line suggestion is based on the plate being

letter, letter, letter, number, number, number.

The letters and numbers could be "mixed-up."

So for the first symbol you have 36 choices (26 letters and 10 digits)

The second also has 36 choices.

And so on, and so on, and so on until 36⁶.

OK?