Question

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

Answer 1

We form the pattern of a number plate that includes two letters at first and then three digits. Since there is no restriction on letters and digits we fill out positions for two letters using total number of letters. Find the total number of license plates that can be formed

• There are 26 letters in English and 10 digits in the number system including 0.
• Fundamental principle of counting is a rule that is used to find the total number of possible outcomes. If we have $x$ ways to do a work and $y$ ways to another work, then the number of ways to do both work is given by $x \times y$.

Complete step-by-step answer:
We have to find a number of license plates that can be made having 2 letters and 3 digits. We calculate total combinations for 2 letters and total combinations for 3 digits separately.
We know alphabets in English are A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y and Z.
$\therefore$Total number of alphabets available$= 26$ … (1)
Also, digits list is 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
$\therefore$Total number of digits available$= 10$ … (2)
To fill the first 2 places we use letters from English alphabets.
$\Rightarrow$Number of ways to fill first 2 positions on the license plate $= 26 \times 26$
$\Rightarrow$Number of ways to fill first 2 positions on the license plate $= 676$ … (3)
To fill last 3 places we use digits
$\Rightarrow$Number of ways to fill last 3 positions on the license plate $= 10 \times 10 \times 10$
$\Rightarrow$Number of ways to fill last 3 positions on the license plate $= 1000$ … (4)
So, we can calculate total number license plates that can be made with 2 letters and then 3 digits using fundamental counting principle.
$\Rightarrow$Total number of license plates $= 676 \times 1000$
$\Rightarrow$Total number of license plates $= 676000$

$\therefore$Total number of license plates that can be formed is 676000.

Note:
Alternate method:
We can solve this question using combination method
We know if we have to choose $r$objects from total $n$objects, then the number of ways to choose objects is given by the formula $^n{C_r}$, where $^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}$
If $r = 1$
${ \Rightarrow ^n}{C_1} = \dfrac{{n!}}{{(n - 1)!1!}}$
Use the formula of factorial i.e. $n! = n \times (n - 1)!$in the numerator
${ \Rightarrow ^n}{C_1} = \dfrac{{n \times (n - 1)!}}{{(n - 1)!1!}}$
Cancel the same terms from numerator and denominator and write $1! = 1$
${ \Rightarrow ^n}{C_1} = n$ … (1)
Since, we are given
Number of total alphabets $= 26$
$\Rightarrow$Number of ways to choose an alphabet ${ = ^{26}}{C_1}$ … (2)
Number of total digits $= 10$
$\Rightarrow$Number of ways to choose a digit $^{10}{C_1}$ … (3)
So, total number of license plates that can be made can be written as
$\Rightarrow$Number of license plates ${ = ^{26}}{C_1}{ \times ^{26}}{C_1}{ \times ^{10}}{C_1}{ \times ^{10}}{C_1}{ \times ^{10}}{C_1}$
Use equation (1) to write the values in the product
$\Rightarrow$Number of license plates $= 26 \times 26 \times 10 \times 10 \times 10$
$\Rightarrow$Number of license plates $= 26000$
$\therefore$Total number of license plates that can be formed is 676000