The letters of the word ORIENTAL are arranged in all possible ways. The chance that the consonants and vowels occur alternatively is
- $\dfrac{1}{70}$
- $\dfrac{2}{35}$
- $\dfrac{1}{35}$
- $\dfrac{4}{35}$
ORIENTAL: 8 letters, 4 vowels (O, I, E, A), 4 consonants (R, N, T, L). Total arrangements = 8!. Favorable: V C V C V C V C or C V C V C V C V. 2 * (4! * 4!) = 2 * 24 * 24 = 1152. Total = 40320. Probability = 1152/40320 = 1/35.
The word ORIENTAL has 8 letters in total, consisting of 4 vowels (O, I, E, A) and 4 consonants (R, N, T, L). The total number of possible arrangements for the letters is 8!, which equals 40320. For the consonants and vowels to occur alternatively, they must occupy either all the even positions and odd positions respectively, or vice versa. In either of the two possible alternating patterns, the 4 vowels can be arranged among themselves in 4! ways, and the 4 consonants can be arranged among themselves in 4! ways. The total number of favorable arrangements is 2 multiplied by 4! multiplied by 4!, which equals 2 multiplied by 24 multiplied by 24, giving 1152. The probability is the ratio of favorable arrangements to total arrangements, which is 1152 divided by 40320, simplifying exactly to 1/35.