Question

In how many distinct ways can the letters of the word STATES be arranged?

Answer 1

Hint: First count the number of letters present in the given word. Even if the letters are repeating then count them a different number of times. Now, use the formula that n number of different things can be arranged in $n!$ ways. Now, divide this total number of ways of arrangement with the product of factorials of the number of times each letter is repeated to get the total number of effective arrangements.

Complete step-by-step solution:
Here we are provided with the word STATES and we are asked the number of possible arrangements that can be made using the letters of this word.
Now, clearly we can see that we have 6 letters in this word if we are counting the repeated letters differently. We know that n things can be arranged in $n!$ ways if there are n positions. So we have,
$\Rightarrow $ Number of ways in which the letters of the given word can be arranged = $6!$
Now, we can see that the letters T and S are repeating twice, so $6!$ would have been the number of possible arrangements in case all the letters would have been different. In cases where we have repetition of letters the effective number of arrangements decreases. The effective number of arrangements will be the number of arrangements obtained in case the letters do not repeat divided by the product of factorials of the number of times each letter is repeating. So we get,
$\Rightarrow $ Effective number of ways in which the letters of the given word can be arranged $=\dfrac{6!}{2!\times 2!}=180$.
Hence, there are 180 arrangements of the letters possible.

Note: Note that here the two letters were repeating two times each and that is why we have taken the product $2!\times 2!$ in the denominator. In case they may repeat n number of times then we have to take $n!$ and products accordingly. Actually here we are applying the concept of permutation where we arrange n things at r places using the formula $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.