Question

counting principle
each license plate in a certain state has six characters (with repeats allowed).
here are the possibilities for each character.
how many license plates are possible in this state?

Explanation:

Step1: Apply the counting principle

The counting principle states that if there are \(m_1\) ways to do the first task, \(m_2\) ways to do the second task, \(\cdots\), \(m_n\) ways to do the \(n\)th task, then the total number of ways to do all \(n\) tasks together is \(m_1\times m_2\times\cdots\times m_n\).
For the license - plate:

• The first character has \(m_1 = 26\) (26 letters) ways.
• The second character has \(m_2=26\) (26 letters) ways.
• The third character has \(m_3 = 26\) (26 letters) ways.
• The fourth character has \(m_4=10\) (10 digits) ways.
• The fifth character has \(m_5 = 10\) (10 digits) ways.
• The sixth character has \(m_6=10\) (10 digits) ways.

Step2: Calculate the product

The total number of license - plates \(N=26\times26\times26\times10\times10\times10\).
We know that \(26\times26\times26=26^{3}=\ 17576\) and \(10\times10\times10 = 10^{3}=1000\).
Then \(N=17576\times1000\).

Answer:

\(17576000\)