Multiple choice

How many different words can be fanned by jumbling the letters of the word MISSISSIPPI in which no two 's' are adjacent

  1. $\displaystyle 7\cdot ^{6}C_{2}\cdot ^{8}C_{4}$
  2. $\displaystyle 6\cdot 7 \cdot ^{8}C_{4}$
  3. $\displaystyle 6\cdot 8 \cdot ^{7}C_{4}$
  4. $\displaystyle 8\cdot ^{6}C_{4}\cdot ^{7}C_{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

MISSISSIPPI has 4 S's, 4 I's, 2 P's, and 1 M. Arrange the 7 non-S letters in 7!/4!2! ways, then select 4 of the 8 gaps around them for the S's so that no two S's are adjacent. This equals 7 x C(6,2) x C(8,4), matching option A.

AI explanation

To ensure no two 'S' letters are adjacent, first arrange the remaining letters: M, I, I, I, I, P, P (7 letters total). These 7 letters create 8 available gaps for the 'S' letters to occupy. We must choose 4 of these 8 gaps for the four 'S' letters, which gives 8C4 ways. The 7 remaining letters can be arranged in 7! / (4! * 2!) ways, which simplifies to (7 * 6 * 5) / 2 or 105 ways, and also equals 7 * 6C2. Multiplying these gives 7 * 6C2 * 8C4.

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