How many distinguishable ways can the letter of the word SASSAFRAS be arranged using all the letters?

Question

Andrew Adams · Accepted Answer

In a word where no letters are repeated, such as FRANCE, the number of distinguishable ways of arranging the letters could be calculated by 5!, which gives 120. However, when letters are repeated, you must use the formula #(n!)/((n_1!)(n_2!)...)#

There are 4 s's, 3 a's and a total of 9 letters.

#(9!)/((4!)(3!))#

= #362880/(24 xx 6)#

= 2520

There are 2520 distinguishable ways of arranging the letters.

Practice exercises:

Find the number of distinguishable ways of arranging the letters in the word EXERCISES.

Find the number of distinguishable ways of arranging letters in the word AARDVARK.

Good luck!

Cameron Adams · Answer

undefined The word "SASSAFRAS" contains 10 letters, with 3 repetitions of 'S', 3 repetitions of 'A', and 2 repetitions of 'F'. Therefore, the number of distinguishable ways to arrange the letters of the word "SASSAFRAS" is given by the formula for permutations of a multiset, which is:

$$ \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} $$

where $ n $ is the total number of letters, and $ n_1, n_2, \ldots, n_k $ are the counts of each distinct letter.

Substituting the values, we get:

$$ \frac{10!}{3! \cdot 3! \cdot 2!} = \frac{3628800}{6 \cdot 6 \cdot 2} = 50400 $$

So, there are 50,400 distinguishable ways to arrange the letters of the word "SASSAFRAS".