Question

How many license plates can be made consisting of $2$ letters followed by $3$ digits (using the fundamental counting principle to solve)?

Answer 1

Hint: First we have to find the number of possibilities for $2$ letters if A is the first letter. Similarly, find the number of possibilities for $2$ letters if B is the first letter. And so on. Then by Fundamental Principle of Multiplication, find the number of arrangements of $2$ letters. Next, find the number of possibilities for $3$ digits. Then by Fundamental Principle of Multiplication, find the number of arrangements of $3$ digits. Then by Fundamental Principle of Addition, find the number of license plates can be made consisting of $2$ letters followed by $3$ digits.

Complete step-by-step solution:
We know there are 26 letters in the alphabet (A, B, C, D,…,Z) and $10$ digits in the number system ($0 - 9$).
In the given statement there is nothing stated that the letters and digits can’t be repeated, so all 26 letters of the alphabet and all $10$ digits can be used again.
First we have to find the number of possibilities for $2$ letters if A is the first letter.
Write all the possibilities possible in this case:
AA, AB, AC, AD, AE,…, AW, AX, AY, AZ.
So, there are $26$ possibilities for $2$ letters if A is the first letter.
Similarly, find the number of possibilities for $2$ letters if B is the first letter.
Write all the possibilities possible in this case.
BA, BB, BC, BD, BE,…, BW, BX, BY, BZ.
So, there are $26$ possibilities for $2$ letters if B is the first letter.
And so on.
So, by Fundamental Principle of Multiplication,
Here there are $2$ letters such that one of them can be completed in $26$ ways, and when it has been completed in any of these $26$ ways, second letter can be completed in $26$ ways; then the two letters in succession can be completed in $26 \times 26$ ways.
Therefore, the number of arrangements of $2$ letters is $676$.
Now, find the number of possibilities for $3$ digits.
The hundred’s place can have any one of the digits from $0$ to $9$. So, hundred’s place can be filled in $10$ ways. The ten’s place can have any one of the digits from $0$ to $9$. So, ten’s place can be filled in $10$ ways. The one’s place can have any one of the digits from $0$ to $9$. So, one’s place can be filled in $10$ ways.
So, we use by Fundamental Principle of Multiplication,
Here there are $3$ digits such that one of them can be completed in $10$ ways, and when it has been completed in any of these $10$ ways, second digit can be completed in $10$ ways, third digit can be completed in $10$ ways; then the $3$ digits in succession can be completed in $10 \times 10 \times 10$ ways.
Therefore, the number of arrangements of $3$ digits is $1000$.
So, we use by Fundamental Principle of Addition,
Here there are two jobs such that they can be performed independently in $676$ and $1000$ ways respectively, then either of the two jobs can be performed in $\left( {676 + 1000} \right)$ ways.

Therefore, the number of license plates can be made consisting of $2$ letters followed by $3$ digits is $1676$.

Note: Fundamental Principles of Counting:
Fundamental Principle of Multiplication: If there are two jobs such that one of them can be completed in $m$ ways, and when it has been completed in any of these $m$ ways, second job can be completed in $n$ ways; then the two jobs in succession can be completed in $m \times n$ ways.
Fundamental Principle of Addition: If there are two jobs such that they can be performed independently in $m$ and $n$ ways respectively, then either of the two jobs can be performed in $\left( {m + n} \right)$ ways.
Here, we can’t use the Fundamental Principle of Multiplication in the last step as there is no common outcome for choosing $2$ letters and $3$ digits.