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How many ways can the word MATHEMATICS be arranged if the last letter must be a T?

My solution:

There are $2$ possible choices for the last letter (There are $2$ different T's), which leaves $10$ choices for other places. However, the letters 'M' and 'A' are repeated twice. Therefore, the answer is $$\frac{10! \times 2}{2! \times 2!}$$

Is this correct?

Aspiring Mathlete
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1 Answers1

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Another way to think of this is the number of arrangements of $$MATHEMAICS$$ (note the missing $T$ wich is reserved for the last letter). The solution now is simply $$\frac{10!}{2!\cdot 2!} = \frac{10!}4$$ due to the same argument as you noted (double $A$ and $M$, all other letters unique).

AlexR
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  • Are there not $2$ choices for the last letter, since there are $2$ T's? – Aspiring Mathlete Apr 27 '15 at 16:30
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    @AspiringMathlete In the new problem there is only one $T$. The old problem is basically equivalent. We "took away" one $T$ from the pool of letters to be arranged. – AlexR Apr 27 '15 at 16:30