Multiple choice

If the letters of the word REGULATIONS are arranged at random, the chances that there will be exactly four letters between R and E is

  1. 9C4 x 4! x 2! x 6!/11!

  2. 9C4/11!

  3. 9C4 x 4! x 2!/11!

  4. 9C4 x 4! x 6!/11!
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A Correct answer
Explanation

Total arrangements of REGULATIONS is 11!. To have 4 letters between R and E, there are 7 possible positions for the pair (R, E). There are 2 ways to arrange R and E (R...E or E...R). The remaining 9 letters can be arranged in 9! ways. The probability is (7 * 2 * 9!) / 11!. This simplifies to 9C4 logic or similar permutation counting.

AI explanation

The total number of ways to arrange the 11 letters of the word REGULATIONS is 11 factorial. To place R and E with exactly four letters between them, the combination 9C4 selects the four intermediate letters, 4 factorial arranges them, and 2 factorial arranges R and E at the ends. The remaining 6 letters are arranged in 6 factorial ways, making the probability equal to 9C4 multiplied by 4 factorial, multiplied by 2 factorial, multiplied by 6 factorial, all divided by 11 factorial.

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