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ABHYAS PREP: ADVANCED QUANT · PERMUTATION & COMBINATION

Arranging letters of a word and people in a circle

Letters with repeats: n! ÷ (p! q! …). Letters that must stay together: tie them into one block, then arrange the block inside. Around a circle: (n − 1)!.

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Worked examples

Example 1 · level 1

A signboard painter in Patna has the letters of MIZORAM cut out of wood. How many different arrangements of all 7 letters are possible?

  1. 360
  2. 2,520
  3. 5,038
  4. 5,040
Show solution
  1. MIZORAM has 7 letters: M twice.
  2. Arrangements = 7! ÷ (2!) = 5,040 ÷ 2.
  3. = 2,520.

Answer: 2,520

Example 2 · level 2

A word-game app made in Guwahati shuffles the letters of GOOGLY. In how many of the arrangements do all the vowels come together?

  1. 24
  2. 60
  3. 180
  4. 240
Show solution
  1. Vowels: O ×2 (2 letters). Consonants: 4 (G twice).
  2. Arrange 5 units (block + consonants): 5! ÷ (2!) = 60.
  3. Arrange the vowels inside the block: 2! ÷ (2!) = 1.
  4. Together: 60 × 1 = 60.

Answer: 60

Example 3 · level 3

5 friends, including Aditya and Aarav, sit around a round table at a Holi get-together in Lucknow. In how many ways can they sit so that Aditya and Aarav are always next to each other?

  1. 6
  2. 12
  3. 24
  4. 48
Show solution
  1. All seatings: (5 − 1)! = 24.
  2. Together: 4 units around the table → 3! = 6; × 2 for Aditya/Aarav order → 12.

Answer: 12

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