Using all the letters of the word POETRY, how many words

Solution

The word POETRY has two vowels – O and E.
If a word starts and ends with a vowel, the two letters to occupy the first and the last positions can be selected and arranged in 2P2 = 2 × 1 = 2 ways.
The remaining 4 letters can be arranged in 4! = 24 ways.
Number of words that start and end with a vowel = 2 × 24 = 48

If a word starts with a vowel but ends with a consonant, its first letter can be selected from O and E = 2 ways.
Last letter can be selected from P, T, R, Y = 4 ways.
Remaining 4 letters arranged in 4! = 24 ways.
So total = 2 × 4 × 24 = 192