How many different license plates can be made if each plate contains a sequence of three alphabets followed by three digits?

How many car number plates can be made if each plate contains 2 different letters of English alphabet, followed by 3 different digits?

Answer

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Hint: There are 26 different letters in English Alphabet and 10 different digits in mathematics. And two letters and the three digits must be different. So the first letter can be selected from 26 letters and the second letter can be selected from the remaining 25 letters. In the same way we have to select the digits too. After finding the no. of ways of selecting letters and digits, multiply both of them together to get the total no. of car number plates can be made. Use combinations.

Complete step-by-step answer:
We are given the no. of car number plates that can be made if each plate contains 2 different letters of English alphabet, followed by 3 different digits.
So the first two terms of the plate are two different letters. There are 26 letters in English alphabet.
First can be selected from 26 letters and the second letter can be chosen from the remaining 25 different letters. Total no. of ways is
 $ \Rightarrow {}_{}^{26}C_1^{} \times {}_{}^{25}C_1^{} $
The last three terms of the plate are three digits. There are 10 different digits in mathematics.
First number can be selected from 10 digits, second from the remaining 9 different digits and the third from the remaining 8 different digits. So the total no. of ways is
 $ \Rightarrow {}_{}^{10}C_1^{} \times {}_{}^9C_1^{} \times {}_{}^8C_1^{} $
Total no. of number plates that can be made is
 $ \Rightarrow {}_{}^{26}C_1^{} \times {}_{}^{25}C_1^{} \times {}_{}^{10}C_1^{} \times {}_{}^9C_1^{} \times {}_{}^8C_1^{} = 26 \times 25 \times 10 \times 9 \times 8 = 468000 $
So, the correct answer is “468000”.

Note: Here we are selecting a letter and number from a group of letters, so we used combinations and the letters and the numbers can be arranged in any order among themselves. When the order does not matter we have to use combinations or else permutations.

How many automobile license plates can be made if each plate contains two different letters followed by three different digits?

Solution

We have 26 English alphabet and 10 digits (0 to 9)

Since, it is given that each plate contains 2 different letters followed by 3 different digits.

∴ Number of arrangement of 26 letters taken 2 at a time

= 26P2

= `(26!)/((26 - 2)!)`

= `(26!)/(24!)`

= `(26*25*24!)/(24!)`

= 650

 Three-digit number can be formed out of 10 digit = 10P3

= `(10!)/(7!)`

= `(10*9*8*7!)/(7!)`

= 720

∴ Total number of license plates = 650 × 720 = 468000.

Concept: Derivation of Formulae and Their Connections

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