12) In how many distinct ways can the letters $L, A, P, I, S$ be arranged so that the first and last letters are vowels?
(A) $6, \quad$ (B) $8, \quad$ (C) 10
(D) 12
(E) 24.
12. The answer is $(\mathbf{D})$.

The word contains exactly 2 vowels that will occupy the first and last positions. This can only be done in 2 ways: "A-- -I" or "I---A". The 3 consonants will be arranged in the 3 central positions; we have 3 choices for the consonant in the second position, 2 (one consonant has already been chosen) for the one in the third position, while for the fourth position the choice is mandatory, as only one letter is left. There are thus $3 \cdot 2 \cdot 1=6$ possible permutations for the three consonants. Remembering that for each possible permutation of the consonants there are 2 possible permutations of the vowels, the total number of arrangements of the letters is $6 \cdot 2=12$.