If the \( n \) objects in a permutations problem are not all distinguishable-that is, if there are \( n_{1} \) of type 1 , \( n_{2} \) of type 2, and so on, for \( r \) different types-then the number of distinguishable permutations is given by \( \frac{n!}{n_{1}!n_{2}!\cdots n_{r}!} \). How many permutations are possible using the 11 letters in the word MISSISSIPPI?

The word "MISSISSIPPI" consists of 11 letters, with the following breakdown by letter type: M (1), I (5), S (4), and P (2). Using the formula for distinguishable permutations, we calculate: \[ \text{Number of permutations} = \frac{11!}{1! \times 5! \times 4! \times 2!} \] Calculating this gives: \[ 11! = 39916800 \] \[ 1! = 1, \quad 5! = 120, \quad 4! = 24, \quad 2! = 2 \] So, \[ 1! \times 5! \times 4! \times 2! = 1 \times 120 \times 24 \times 2 = 5760 \] Now divide: \[ \frac{39916800}{5760} = 6930 \] Thus, there are 6,930 distinguishable permutations of the letters in the word "MISSISSIPPI". Did you know that the concept of permutations dates back to ancient mathematicians? In fact, early notions of arranging objects can be traced to Hindu scholars who studied combinatorial problems. It was a fascinating blend of practical problem-solving and pure mathematical explorations! Speaking of permutations, a common mistake is forgetting to account for indistinguishable objects or miscounting them. Always check that you're accurately identifying and grouping identical items when applying the permutation formula—it can make a huge difference in your final total!