Question

You are creating a license plate for your new car. The license plate must start with 3letters followed by 4 single-digit numbers, without any repetition of letters or numbers.How many options do you have?

Answer 1

Given:

The license plate must start with 3 letters followed by 4 single-digit numbers without any repetition of numbers and letters.

Let's find the number of options we have to create the license plate for the new car.

To find the number of options, we have:

Total number of letters( a to z) = 26

Total numbers (0 to 9) = 10

We are to pick 3 letters from 26 letters

Also, we are to choose 4 digits from 10 digits

To find the number of options, we are to use combination method.

Thus, we have:

`^(26)C_3\ast^(10)C_4`

Solving further:

`\begin{gathered} (26!)/((3!(26-3)!))\ast(10!)/((4!(10-4)!)) \\ \\ (26!)/((3!(23)!))\ast(10!)/((4!(6)!)) \\ \\ (26\ast25\ast24\ast23!)/(3\ast2\ast1\ast23!)\ast(10\ast9\ast8\ast7\ast6!)/(4\ast3\ast2\ast1\ast6!) \\ \\ (26\ast25\ast24)/(3\ast2\ast1)\ast(10\ast9\ast8\ast7)/(4\ast3\ast2\ast1) \end{gathered}`

Solving further, we have:

`\begin{gathered} (15600)/(6)\ast(5040)/(24) \\ \\ 2600\ast210 \\ \\ =546000 \end{gathered}`

Therefore, we have 546000 options to choose from.

ANSWER:

546,000 options