[tex] \large\underline \mathcal{{QUESTION:}}[/tex]
In how many different ways can be letter of the word RUMOUR be arranged if the consonants always come together?
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[tex] \large\underline \mathcal{{SOLUTION:}}[/tex]
Consonants = RMR = 1 entity
Other letters = UOU = 3 entities
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Now , There are 4 entities. In here we will solve for the distinguishable permutation. Because UOU , the letter U is repeated twice. And thesame goes for the Consonants:
- Total entities = 3+1 = 4
- Repeated Other letters = 2 (U)
- Total Consonants = RMR = 3
- Repeated Consonant = 2 (R)
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[tex]\sf{P=\frac{(Total\:entities)!}{(Repeated(U))!}\times\frac{(Total\:Consonants)!}{(Repeated(R))!}}[/tex]
[tex]\sf{P=\frac{4!}{2!}\times\frac{3!}{2!}}[/tex]
[tex]\sf{P=\frac{4\times3\times2\times1}{2\times1}\times\frac{3\times2\times1}{2\times1}}[/tex]
[tex]\sf{P=\frac{4\times3\times\cancel{2\times1}}{\cancel{2\times1}}\times\frac{3\times\cancel{2\times1}}{\cancel{2\times1}}}[/tex]
[tex]\sf{P=4\times3\times3}[/tex]
[tex]\sf{P=36}[/tex]
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[tex] \large\underline \mathcal{{ANSWER:}}[/tex]
