A code word is to consist of two distinct English alphabets followed by two distinct number from `1to9.` for example, `C A23` is a code word. How many such code words are there? How many of them end with an even integer?

Explanation

(i) We know that there are 26 letters in the English alphabet.
number of ways of choosing 2 letters out of 26
`=""^(26)P_(2)=(26xx25)=650.`
Number of ways of choosing 2 numbers out of 9
`=""^(9) P_(2)=(9xx8)=72.`
By the fundamental principle of multiplication, the total number of codes `=(650xx72)=46800.`
(ii) Particular case When each code ends with an even integer
In this case, it can have any of the numbers 2, 4, 6, 8 at the extreme right position. This position can thus be filled in 4 ways.
The next left position can be filled with any of the remaining 8 numbers. So, there are 8 ways of filling this position.
Number of ways of choosing pairs of even numbers
`=(4xx8)=32.`
Number of ways of choosing 2 letters out of 26
`=""^(26)P_(2)=(26xx25)=650.`
Hence , the number of such codes which end with an even
Integer `=(32xx650)=20800.`