Verified answer

[tex]\boxed{210}[/tex]

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Stepwise Explanation

Permutation with Repetition

The arrangements when using all the limited letters.

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Letters Contained

[tex]\begin{array}{| c | c | c | c |} \cline{1-4} \textrm{R} & \textrm{S} & \textrm{T} & \textrm{Total} \\ \cline{1-4} \textrm{3} & \textrm{2} & \textrm{2} & \textrm{7}\\ \cline{1-4} \end{array}[/tex]

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Consider [tex]\text{$\rm 7!$}[/tex]

It is the number of ways to arrange seven distinct letters.

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Consider [tex]\text{$\rm 3!$}[/tex]

It is the number of ways to arrange [tex]\text{$\rm {R}_{1}$, $\rm {R}_{2}$, $\rm {R}_{3}$}[/tex].

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Consider [tex]\text{$\rm 2!$}[/tex]

It is the number of ways to arrange [tex]\text{$\rm {S}_{1}$, $\rm {S}_{2}$}[/tex].

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Consider [tex]\text{$\rm 2!$}[/tex]

It is the number of ways to arrange [tex]\text{$\rm {T}_{1}$, $\rm {T}_{2}$}[/tex].

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In the arrangement, the hierarchy of the same letters doesn't exist. We should divide not to overcount the same letters.

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Hence,

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[tex]\text{$\dfrac{7!}{3!2!2!}$}[/tex]

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[tex]\text{$=\dfrac{7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{6\cdot2\cdot2}$}[/tex]

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[tex]\text{$=7\cdot5\cdot3\cdot2\cdot1$}[/tex]

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[tex]\text{$=210$}[/tex]

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[tex]\text{\boxed{210} ways exist.}[/tex]

Answer:

210 different words can be formed when the letter R is used thrice and letters S and T are used twice each.

Step-by-step explanation: