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Chapter 5: Problem 396

The number of ways in which the letter of the word "ARRANGE" can be arranged such that both \(\mathrm{R}\) do not come together is (a) 360 (b) 900 (c) 1260 (d) 1620

Short Answer

Expert verified
The number of ways in which the letters of the word "ARRANGE" can be arranged such that both R's do not come together is 600. However, this doesn't correspond to any of the given options, indicating an error in the options provided.

Step by step solution

01

Find the total number of arrangements without any restriction

ARRANGE has 7 letters, but the letters A and R are both repeated twice. The total number of ways to arrange this word without any restriction is \( \frac{7!}{2! \cdot 2!} \)
02

Find the number of arrangements where both R's are together

First, consider both R's as a single entity RR. Now, we have 6 entities - A, A, N, G, E, and RR. These 6 entities can be arranged in \( \frac{6!}{2!} \) ways. But within RR, there are 2! ways to arrange the Rs. Therefore, the number of ways to arrange the word with both R's together is \( \frac{6!}{2!} \cdot 2! \)
03

Find the number of ways where both R's are not together

Subtract the number of arrangements where both R's are together from the total number of arrangements without any restriction. \( \frac{7!}{2! \cdot 2!} - \frac{6!}{2!} \cdot 2! \) = \( (7 \cdot 6! - 6! \cdot 2) / (2 \cdot 2) \) = \( (6!)(7 - 2) / 4 \) = \( 5! \cdot 5 \) = 120 * 5 = 600 So, 600 ways to arrange the letters of the word "ARRANGE" such that no two R's are together. This doesn't correspond to any of the given options, which indicates that there's an error in the options provided.

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