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
$=26^{4}-21^{4}-5^{4}$
$=(26^{2}+21^{2})(26^{2}-21^{2})-5^{4}$
$=(676+441) (5) (47)-5^{4}$
$=5[(1117\times 47)-125]$
$=5 \times (52374)$
$=261870$