if letters of the word REGULATIONS are arranged at random. what is the probability that there will be exactly 4 letters between R and E?
To find the probability of exactly 4 letters between R and E in the word REGULATIONS when the letters are arranged randomly, we need to determine the total number of possible arrangements and the number of arrangements that meet the given condition.
1. Total number of arrangements: The word "REGULATIONS" has 11 letters.
Therefore, the total number of arrangements is 11!, which can be calculated as 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 39,916,800.
2. Number of arrangements with exactly 4 letters between R and E:
Let's consider "RE" as a single unit. We can arrange the remaining 9 letters (GULATIONS) in the remaining 9 positions. However, the "RE" unit can be arranged in two different ways (RE or ER).
Therefore, the number of arrangements with exactly 4 letters between R and E is 2 × 9! = 725,760.
3. Probability: The probability is given by the number of arrangements that meet the given condition divided by the total number of arrangements.
Probability = Number of arrangements with exactly 4 letters between R and E / Total number of arrangements
Probability = 725,760 / 39,916,800
Therefore, the probability of finding exactly 4 letters between R and E when the letters of the word REGULATIONS are arranged randomly is approximately 0.0182 or 1.82%.