Numerical answer
In how many ways can the letters of the word ORANGE be arranged so that all three vowels occupy only the odd positions (1st, 3rd and 5th)?
Show answer and solution
Answer: 36
ORANGE has 6 distinct letters: vowels O, A, E and consonants R, N, G.
The odd positions (1, 3, 5) must hold exactly the 3 vowels, and the even positions (2, 4, 6) must hold the 3 consonants.
Arrange the 3 vowels in the 3 odd slots: 3! = 6 ways.
Arrange the 3 consonants in the 3 even slots: 3! = 6 ways.
Total = 6 × 6 = 36.
Tip: When a restriction splits the positions into two fixed groups (odd/even, vowel/consonant slots), arrange each group independently and multiply.
The odd positions (1, 3, 5) must hold exactly the 3 vowels, and the even positions (2, 4, 6) must hold the 3 consonants.
Arrange the 3 vowels in the 3 odd slots: 3! = 6 ways.
Arrange the 3 consonants in the 3 even slots: 3! = 6 ways.
Total = 6 × 6 = 36.
Tip: When a restriction splits the positions into two fixed groups (odd/even, vowel/consonant slots), arrange each group independently and multiply.
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