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 word ARTICLE has 7 distinct letters, so the total number of possible arrangements is 7!. There are 3 vowels (A, I, E) and 4 consonants (R, T, C, L). To find the favorable arrangements, choose 1 of the 3 vowels for the first position and 1 of the 4 consonants for the last position. The remaining 5 letters can be arranged in the middle in 5! ways. The probability is the ratio of favorable to total arrangements: (3 times 4 times 5!) / 7!. Simplifying the fraction 7! to 7 times 6 times 5! gives 12 / 42, which reduces to 2/7.

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