Question

Find the number of different ways of
arranging letters in the word ARRANGE.
How many of these arrangements the two
R's and two A's are not together?​

Answer 1

Answer:

Now, in our question, we have word ARRANGE where total 7 words are there out of which 2 are A's and 2 are R's and rest 3 are different then, the total number of ways in which letters of the given word can be arranged will be 7! (2!) (2!) =1260 .

Step-by-step explanation:

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Answer 2

Answer:

Number of possible arrangements = 1260

Number of arrangements where two R's and A's do not come together = 660

Step-by-step explanation:

Given:

• The word ARRANGE

To Find:

• The number of different ways of arranging the letters of the given word
• The number of arrangements possible if the two A's and R's are not together

Solution:

Here the letters R and A repeat two times.

Hence the number of possible arrangements of the letters of the word ARRANGE is given by,

$\sf Number\:of\:ways=\dfrac{7!}{2!\times 2!}$

$\implies \sf \dfrac{7\times 6\times 5\times 4\times 3\times 2}{2\times 2}$

$\implies \sf 7\times 6\times 5\times 3\times 2$

$\implies \sf 1260$

Hence the letters of the word ARRANGE can be arranged in 1260 different ways.

Now finding the number of arrangements where the two R's and two A's are not together,

First find the number of arrangements where the two A's come together.

Considering the two A's as a single letter,

$\sf Number\:of\:ways=\dfrac{6!}{2!}$

$\sf \implies 360$

Finding the number of ways in which the two R's come together,

$\sf Number\:of\:ways=\dfrac{6!}{2!}=360$

Finding the number of ways where both two A's and R's come together,

Number of ways = 5! = 120

Now number of ways in which the two R's and A's do not come together is given by,

Number of ways = 1260 - (360 + 360 - 120)

= 1260 - 600

= 660

Hence there are 660 number of possible ways where the two A's and R's do not come together.