Text Twist
In the game Text Twist, six letters are given and the player must form words of different lengths using those letters. Suppose the letters in a particular game are ENHSIC.


c. The solution to this game has three 6-letter words. To move to the next round, the player must get at least one of the six-letter words. If the player simply guesses, what is the probability of getting one of the six-letter words on the first try?

Verified step by step guidance

1. Identify the total number of possible 6-letter arrangements (permutations) that can be formed using the letters ENHSIC. Since all letters are distinct, the total number of permutations is given by the factorial of 6: \$6!$.
2. Calculate the total number of favorable outcomes. According to the problem, there are exactly three valid 6-letter words that can be formed from these letters. So, the number of favorable outcomes is 3.
3. Recall that the probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability of guessing one of the six-letter words on the first try is \(\frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\).
4. Substitute the values into the probability formula: \(\frac{3}{6!}\).
5. Interpret the result as the chance of randomly selecting one of the three valid six-letter words from all possible permutations of the letters ENHSIC.

Key Concepts

Permutation and Counting Outcomes

Permutation refers to the arrangement of all members of a set into some sequence or order. In this problem, the total number of possible 6-letter words formed from the letters ENHSIC is the number of permutations of 6 distinct letters, calculated as 6! = 720. This represents the total number of equally likely outcomes when guessing a word.

Favorable Outcomes

Favorable outcomes are the specific outcomes that satisfy the event of interest. Here, the favorable outcomes are the three valid 6-letter words that can be formed from the letters ENHSIC. Identifying the number of favorable outcomes is essential to calculate the probability of guessing one correctly.

Probability of an Event

Probability measures the likelihood of an event occurring and is calculated as the ratio of favorable outcomes to total possible outcomes. For guessing a six-letter word correctly, the probability is the number of valid six-letter words divided by the total permutations (3/720). This quantifies the chance of success on a random guess.