Calculating the Probability of Exactly Two Letters Repeating in a 6-Letter Word with Replacement

When forming a 6-letter word from a set of 16 different letters with replacement, we want to understand the probability that exactly two letters are repeated. This is a classic combinatorial problem that involves multiple steps, including selection and arrangement. Let's explore the detailed solution step by step.

Step 1: Choosing the Letters

In this problem, we need to choose which letters will be repeated and which will be unique. This involves two main choices:

Choosing 2 Letters to Repeat

The first step is to choose which two letters will be the ones that are repeated. We can do this in:

[ binom{16}{2} frac{16 times 15}{2} 120 ]

ways, since we are choosing 2 different letters out of 16 without regard to order.

Choosing 4 More Letters

After selecting the two letters to repeat, we need to choose 4 more different letters from the remaining 14 letters. The number of ways to choose these 4 letters is:

[ binom{14}{4} frac{14!}{4!(14-4)!} frac{14 times 13 times 12 times 11}{4 times 3 times 2 times 1} 1001 ]

This ensures that the same letter is not chosen more than once.

Step 2: Arranging the Letters

The next step is to arrange the 6 letters where 2 letters are repeated. The number of ways to arrange these 6 letters is given by the multinomial coefficient:

[ frac{6!}{2! cdot 4!} frac{720}{2 cdot 24} 15 ]

This accounts for the fact that 2 letters are identical, and the remaining 4 are distinct.

Step 3: Calculating the Number of Favorable Outcomes

Now we multiply the number of ways to choose the letters by the number of ways to arrange them:

[ text{Favorable outcomes} binom{16}{2} times binom{14}{4} times frac{6!}{2! cdot 4!} 120 times 1001 times 15 ]

This gives us:

[ 1201200 ]

Step 4: Total Possible Outcomes

The total number of possible 6-letter words that can be formed from 16 letters with replacement is:

[ 16^6 16777216 ]

Step 5: Calculating the Probability

The probability that exactly two letters are repeated is the ratio of the number of favorable outcomes to the total number of possible outcomes:

[ P frac{1801800}{16777216} approx 0.1074 ]

This can also be expressed as a percentage:

[ P approx 10.74% ]

Thus, the probability that exactly two letters are repeated in a 6-letter word formed from a set of 16 different letters with replacement is approximately 0.1074 or 10.74%.

Key Takeaways

Probability: The likelihood of an event occurring. Combinatorics: The branch of mathematics dealing with combinations of objects and their properties. Binomial Coefficient: Used to count the number of ways to choose a subset of items from a larger set. Multinomial Coefficient: Used to count the number of ways to arrange items when some items are indistinguishable.

Conclusion

Understanding the steps involved in calculating the probability of repeated letters in a word is crucial in fields like statistics, probability theory, and data science. This method not only provides the probability but also helps in understanding the underlying combinatorial principles.