Multiple choice

The letters of the word "$\text{QUESTION}$" are arranged in a row at random. The probability that there are exactly two letters between $Q$ and $S$ is

  1. $\dfrac{1}{14}$
  2. $\dfrac{5}{7}$
  3. $\dfrac{1}{7}$
  4. $\dfrac{5}{28}$
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D Correct answer
Explanation

Total arrangements of QUESTION is 8! = 40320. To have exactly two letters between Q and S, treat them as a block with two letters in between. There are 6 possible positions for the Q-x-x-S block. For each, there are 2! ways to arrange Q and S, and 6! ways to arrange the remaining letters. Total = 6 * 2 * 6! = 8640. Probability = 8640 / 40320 = 5/28.

AI explanation

The total number of ways to arrange the 8 letters of the word QUESTION is 8 factorial. To find the favorable outcomes where there are exactly two letters between Q and S, we first treat Q, the two middle letters, and S as a single block of 4 letters. This block can be chosen and placed in 5 different positions within the 8 spots, Q and S can be arranged in 2 ways within the block, the two middle letters can be chosen in 6 permutation 2 ways, and the remaining 4 letters can be arranged in 4 factorial ways. The probability is then (5 times 2 times 6 permutation 2 times 4 factorial) divided by 8 factorial, which simplifies to 5 divided by 28. The result is 5/28.

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