In how many ways can the letter of the word "ARRANGE" be arranged in which two Rs and two As come together?
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1Your prior question was closed because you did not include any of your efforts. What have you tried? Where are you getting stuck? – lulu Oct 19 '18 at 11:35
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1It is a Permutation with repetitions – Dr. Sonnhard Graubner Oct 19 '18 at 11:37
1 Answers
5
We have to arrange all the letters of word ARRANGE such that AA and RR come together.
So, Put both A's and R's together,
i.e. consider AA and RR as single entities.
n items can be arranged in n! ways:
Proof:
$1^{st}$ item has n options, $2^{nd}$ item has (n-1) options,....$n^{th}$ item has 1 option.
So, total ways are:$$n\cdot(n-1)\cdot(n-2)\cdot\cdot\cdot1$$ i.e. n! ways.
You have AA,RR,N,G,E i.e. 5 entities to be arranged.
You can arrange 5 items amongst themselves in 5! ways.
So, final answer is 5!=120 ways.
pooja somani
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But AA and RR are not always next to each other so how can you treat them like single entries? – Eleven-Eleven Oct 19 '18 at 11:46
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Whats wrong to consider AA and RR as a single entity? That gives 5! – pooja somani Oct 19 '18 at 11:49
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@omega your approach is correct (+1). In permutations with repetition are allowed the (numerous) cases where A,A or R,R are not neighbours. – user376343 Oct 19 '18 at 12:04