Problem 59 - Algebra for College Students

Question

Each arrangement of the seven letters of the word OSMOSIS is put on a slip of paper and placed in a hat. One slip is drawn at random from the hat. Find the probability that the slip contains an arrangement of the letters with an \(\mathrm{O}\) at the beginning and an \(\mathrm{O}\) at the end. \(\frac{1}{21}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{21}\).

Step 1: Calculate Total Arrangements

First, determine the total number of arrangements of the word 'OSMOSIS'. The word has 7 letters where 'S' appears three times and 'O' two times. The formula for permutations of a multiset is given by \( \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} \) where \( n! \) is the factorial of the total number of items and \( n_1! \) to \( n_k! \) are factorials of the frequencies of the repeated items. Thus, the total arrangements can be calculated as: \[ \frac{7!}{2! \cdot 3!} = \frac{5040}{12} = 420 \]

Step 2: Calculate Arrangements with 'O' at Beginning and End

Next, calculate the number of arrangements with 'O' at the beginning and end. Fix 'O' at the first and last position, leaving 5 positions to fill with 'SMOSIS'. Here, 'S' appears three times. Use the multiset formula again: \[ \frac{5!}{3!} = \frac{120}{6} = 20 \]

Step 3: Calculate Probability

Finally, find the probability of drawing a slip with 'O' at the beginning and end by dividing the number of favorable outcomes by the total number of arrangements. Use the formula: \[ \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{20}{420} = \frac{1}{21} \]

Step 4: Conclusion

The probability that a randomly drawn slip contains an arrangement with 'O' at the beginning and end is \(\frac{1}{21}\).

Key Concepts

Permutations

Permutations are all about arranging items in a specific order, where each arrangement or ordering is unique. When dealing with permutations, especially in problems like arranging the letters of a word, the order is critical.
Here's a simple example to help illustrate. If you have three letters, A, B, and C, the permutations would include ABC, ACB, BAC, BCA, CAB, and CBA. Notice how changing the order results in a completely different permutation.
In the context of the problem with the word 'OSMOSIS', permutations help us determine the total number of unique ways to arrange these letters, taking into account repetitions (like the three S's and two O's). Each different arrangement is a different permutation, and calculating permutations of a word helps in figuring out probabilities for specific configurations.

Factorial

Factorials are a mathematical operation that you often encounter when working with permutations and combinations. A factorial is defined as the product of an integer and all the integers below it, down to 1.
For instance, the factorial of 5, denoted as 5!, is calculated as 5 x 4 x 3 x 2 x 1 = 120. Factorials grow very quickly as the number increases.
In the problem, factorials played a crucial role in calculating both the total number of permutations and those specific ones where 'O' appears at the beginning and end. The formula for permutations of a multiset, involving factorials, allows us to adjust our calculations to take repeated items into account. For 'OSMOSIS', the calculation used was \( \frac{7!}{2! \cdot 3!} \), beautifully facilitating the solution by adjusting for the repeated 'S' and 'O'.
Keep in mind, understanding how to compute factorials is essential when dealing with these types of probability problems.

Multisets

Multisets are like sets but allow for multiple occurrences of the same element. This is essential when dealing with words or groups where items repeat.
Think of a multiset in a word such as 'OSMOSIS', where some letters repeat. The multiset for this word would account for both the three 'S's and two 'O's.
When calculating permutations of a multiset, you can't treat repeated items as distinct, which is why the formula for permutation of a multiset \( \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} \) is used. This formula compensates for the repetition by dividing by factorials of the counts of repeated items, helping us arrive at the accurate count of unique permutations.

Combinatorics

Combinatorics is the broad field of mathematics that studies counting, arrangement, and combination of objects. It's at the heart of this probability problem.
Combinatorics is what enables us to determine exactly how many permutations are possible for the word 'OSMOSIS', and more specifically, how many align with the specific criteria of having 'O' at the start and end.
This field uses tools like permutations, combinations, factorials, and the formula for multisets to solve complex counting problems efficiently. In the context of determining probabilities, combinatorics provides powerful methods to enumerate possibilities and thereby calculate the likelihood of specific outcomes occurring in an event, like drawing a word 'OSMOSIS' with specific conditions. Understanding combinatorics' principles injects clarity and precision into solving these probability conundrums.