Question

A code word consists of three letters of the English alphabet followed by two digits of the decimal system. If neither letter nor digit is repeated in any code word, then the total number of code words is:
A. 1404000
B. 16848000
C. 2808000
D. 157010

Answer 1

Hint: A code word consists of three letters followed by two digits. There are 26 letters in the alphabet and 10 digits in the decimal system. Use combinations to find the no. of code words that can be formed.

Complete step-by-step answer:
There are 26 letters in the English Alphabet i.e. 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 Z.
There are 10 digits in the decimal system i.e. 0 1 2 3 4 5 6 7 8 9.
So the first place of the code word can be chosen in 26 ways. Since no letter must be repeated the second letter can be chosen in 25 ways and the third letter can be chosen in 24 ways.
In the same way, the first digit can be chosen in 10 ways and the second digit can be chosen in 9 ways since no digit must be repeated.
Here there is no order to be followed, we are just selecting the letters and digits. So use combinations.
Therefore, total no. of code words that can be formed is
 $
\Rightarrow {}^{26}{C_1} \times {}^{25}{C_1} \times {}^{24}{C_1} \times {}^{10}{C_1} \times {}^9{C_1} \\
\Rightarrow {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} \\
\Rightarrow {}^{26}{C_1} = 26,{}^{25}{C_1} = 25,{}^{24}{C_1} = 24,{}^{10}{C_1} = 10,{}^9{C_1} = 9 \\ $
 $ \Rightarrow No.of words = 26 \times 25 \times 24 \times 10 \times 9 = 1404000 \\
 $
So, the correct answer is “Option A”.

Note: A Permutation is arranging the objects in order. Combinations are the way of selecting the objects from a group of objects or collection. When the order of the objects does not matter then it should be considered as Combination and when the order matters then it should be considered as Permutation. Do not confuse using a combination, when required, instead of a permutation and vice-versa.