At random all the letters of the word "Article" are arranged in all possible ways. The probability that the arrangement begins with vowel and ends with a consonant is

The number of letters in the word ARTICLE is 7

Hence, the number of ways of arranging the letters among themselves without any restriction is 7!

Therefore,   $n\left(S\right)=7!$ 

The number of vowels is 3, and the number of consonants is 4

suppose that E be the event of getting words which starts with vowel and ends with consonants 

$n\left(E\right)=3_{C_1}\times 4_{C_1}\times 5!$

$P\left(E\right)=\frac{3\times 4\times 5!}{7!}=\frac{2}{7}$

The probability that the arrangement begins with vowel and ends with a consonant is  $\frac{2}{7}$