Question

Let $E$ denote the set of letters of the English alphabet, V={$a,e,i,o,u$} and $C$ be the complement of $V$ in $E$. Then, the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is

• A. 261870
• B. 3144160
• C. 425880
• D. 851760

Answer 1

Answer: (A)

The number of all four-letter words $=26^4$
The number of all four-letter words from set $V=\{a$.e. $i.0,u\}$ i.e vowels $=5^4$
Similarly, the number of all four-letter words from set of consonants $=21^5$
$\therefore$ Number of words which contains atleast one vowel and atleast one consonant
$=26^4-21^4-5^4$
$=(26^2+21^2)(26^2-21^2)-5^4$
$=(676+441)(5)(47)-5^4$
$=5[(1117\times 47)-125]$
$=5\times (52374)$
$=261870$