The number of all four-letter words $={26}^4$

The number of all four-letter words from set $V=\left\{a,e,i,o,u\right\}$ 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

$={26}^4-{21}^4-5^4$

$=\left({26}^2+{21}^2\right)\left({26}^2-{21}^2\right)-5^4$

$=\left(676+441\right)\left(5\right)\left(47\right)-5^4$

$=5\left[\left(1117\times 47\right)-125\right]$

$=5\times \left(52374\right)=261870$