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profileLunapopup
asked 2020-02-28 18:00:59 
How many ways can the letters of the word 'ARRANGEMENT' be arranged so that there exists exactly two pairs of consecutive identical letters.
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profiledev
answered 2020-03-05 00:59:56 

You can use this tool for practicing questions like this. Enter the word 'ARRANGEMENT' and generate questions using the tool mentioned. See solutions of generated questions, especially the $24\text{th}$ question. Using similar ways, this problem can also be solved.

The word 'ARRANGEMENT' has $11$ letters where 'E' occurs $2$ times, 'N' occurs $2$ times, 'R' occurs $2$ times and 'A' occurs $2$ times.

Group both 'A's together and consider as a single letter. Similarly, group both 'E's together and consider as a single letter. Then there are $9$ letters in which 'N' occurs $2$ times and 'R' occurs $2$ times. Therefore, number of arrangements where both 'A's are together and both 'E's are together, $n(ae)=\dfrac{9!}{2!×2!}$

We can select any two repeating letters out of four and do the same procedure as above. therefore, $n(ae)+n(an)+n(ar)+n(en)+n(er)+n(nr)=4\text{C}2×\dfrac{9!}{2!×2!}=544320$

Now group both 'A's together, both 'E's together, and both 'N's together. So we have $8$ letters in which 'R' appears two times. Therefore, number of arrangements where both 'A's are together and both 'E's are together and both 'N's are together, $n(aen)=\dfrac{8!}{2!}$

Since we can select any $3$ letters out of $4$,
$n(aen)+n(aer)+n(anr)+n(enr)=4\text{C}3×\dfrac{8!}{2!}=80640$

Now group both 'A's together, both 'E's together, both 'N's together and both 'R's together. So we have $7$ letters and therefore number of arrangements where both 'A's are together and both 'E's are together and both 'N's are together and both 'R's are together, $n(aenr)=7!=5040$

Therefore, using inclusion-principle, number of arrangements where exactly two pairs of consecutive identical letters exist
$=[n(ae)+n(an)+n(ar)+n(en)+n(er)+n(nr)]-3[n(aen)+n(aer)+n(anr)+n(enr)]+6[n(aenr)]\\=544320-3(80640)+6(5040)=332640$

Note: To know more about such problems and general formulas, go through problems related to inclusion-exclusion, especially dealing with $4$ ore more conditions and cases where we count elements matching exactly/at least certain conditions.

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