How Many Ways Can the Letters of the Word 'Statistics' Be Arranged to Begin with a Vowel?

Understanding how many different ways the letters of the word 'statistics' can be arranged such that it begins with a vowel is a fascinating problem in combinatorial mathematics. This involves fundamental principles of permutations and frequency analysis. Let's delve into the detailed steps to find the solution.

Word Breakdown and Frequency Analysis

The word 'statistics' contains several repeated letters. Here’s the frequency analysis of each letter:

S: 3 T: 3 A: 1 I: 2 C: 1

Among these letters, 'a' and 'i' are the vowels. Therefore, we need to consider the arrangements of the letters starting with either 'a' or 'i'.

Case 1: Arrangements Starting with 'a'

When the word begins with 'a', the remaining letters are: s t t i s t i c s. We now calculate the permutations of these letters, taking into account the frequency of each distinct letter.

The formula for permutations of a multiset is given by:

text{Total arrangements} 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 frequencies of each distinct letter.

Calculation for 'a' Case

Total letters left: 9 Frequencies: s(3), t(3), i(2), c(1)

begin{aligned} text{Total arrangements} frac{9!}{3! cdot 3! cdot 2! cdot 1!} frac{362880}{6 cdot 6 cdot 2 cdot 1} frac{362880}{72} 5040 end{aligned}

Case 2: Arrangements Starting with 'i'

When the word begins with 'i', the remaining letters are: s t a t s t i c s. We again use the multiset permutation formula to find the total arrangements.

Calculation for 'i' Case

Total letters left: 9 Frequencies: s(3), t(3), a(1), i(1), c(1)

begin{aligned} text{Total arrangements} frac{9!}{3! cdot 3! cdot 1! cdot 1! cdot 1!} frac{362880}{6 cdot 6 cdot 1 cdot 1 cdot 1} frac{362880}{36} 10080 end{aligned}

Final Calculation

To find the total number of ways the word 'statistics' can be arranged starting with a vowel, we simply add the results from both cases:

text{Total arrangements starting with a vowel} 5040 10080 15120

This means there are 15,120 distinct arrangements of the letters in 'statistics' where the word begins with a vowel.

General Considerations

For a broader perspective, consider the total number of permutations of the word 'statistics' without any restrictions. The formula for this is given as:

text{Total possible permutations} frac{10!}{3!3!2!1!1!1!} 50400

Each letter in 'statistics' can start a permutation, and each permutation can be counted as occurring 5040 times if we total them all. Since there are 3 vowels (a, i, i), the total permutations starting with a vowel would be:

text{Total permutations starting with a vowel} 5040 times 3 15120

Conclusion

Understanding permutations and analyzing the frequency of elements in a set is crucial in solving such combinatorial problems. The word 'statistics' provides a practical example of how to apply these principles to find the number of distinct arrangements beginning with a vowel.