Two letters always together — Aptitude Question
Two letters always together: In how many ways can the letters of ORANGE be arranged so that the two vowels O and A… Answer and worked solution on CodeKairo.
- Topic: Permutations & Combinations
- Section: Quantitative Aptitude
- Difficulty: Medium
- Time target: 90 seconds
Question
In how many ways can the letters of ORANGE be arranged so that the two vowels O and A are always together?
Options
- 120
- 240
- 360
- 480
Show the answer and the worked solution
Correct answer: B. 240
Worked solution
- Treating O and A as one block gives 5 items: [OA], R, N, G, E.
- These arrange in 5! = 120 ways.
- Within the block, O and A can swap: 2 ways.
- Total = 120 × 2 = 240.
Answer: 240.
The idea
Glue the letters that must stay together into a single unit, arrange everything, then multiply by the internal arrangements of the unit.
Shortcut. O and A fill 2 of 6 positions, and 5 of the C(6,2) = 15 position pairs are adjacent, so one arrangement in three keeps them together: 720 ÷ 3 = 240.
Common trap. "120" — gluing O and A into one block and arranging 5! = 120 units, but forgetting that the block can read OA or AO.
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