Understanding Distinguishable Permutations

When we talk about distinguishable permutations, we are looking at the different ways to arrange the letters in a word, where identical letters are considered the same. For example, in the word BANANA, swapping the first 'A' with the second 'A' does not create a new, distinguishable arrangement because the letters are identical.

Analyzing the Letters in BANANA

The word BANANA has a total of 6 letters. Let's count how many times each unique letter appears:

So, we have 1 'B', 3 'A's, and 2 'N's, for a total of $1 + 3 + 2 = 6$ letters.

Formula for Distinguishable Permutations

To find the number of distinguishable permutations of a set of objects where some objects are identical, we use the following formula:

Number of permutations = $\frac{n!}{n_1! n_2! \dots n_k!}$

Where:

• $n$ is the total number of objects (letters in the word).
• $n_1, n_2, \dots, n_k$ are the frequencies of each unique type of object (each unique letter).

Calculating Permutations for BANANA

Using the formula with the letter counts from BANANA:

• Total letters, $n = 6$.
• Frequency of 'A', $n_1 = 3$.
• Frequency of 'N', $n_2 = 2$.
• Frequency of 'B', $n_3 = 1$.

Substitute these values into the formula:

Number of permutations = $\frac{6!}{3! 2! 1!}$

Now, let's calculate the factorial values:

• $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
• $3! = 3 \times 2 \times 1 = 6$
• $2! = 2 \times 1 = 2$
• $1! = 1$

Plug these values back into the formula:

Number of permutations = $\frac{720}{6 \times 2 \times 1}$

Number of permutations = $\frac{720}{12}$

Finally, perform the division:

Number of permutations = $60$

Final Answer for BANANA Permutations

There are 60 distinguishable permutations of the letters in the word BANANA.

Revision Table: Permutation Formulas

Additional Information: Understanding Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, dealing with counting arrangements and selections of objects.

• Permutations: These are arrangements of objects where the order matters. For example, arranging books on a shelf is a permutation because the order of the books creates a different arrangement. The problem with BANANA is a permutation because the order of letters forms a distinct word arrangement.
• Combinations: These are selections of objects where the order does not matter. For example, choosing a team of 3 players from a group of 10 is a combination because the order in which you pick the players does not change the team itself.

The key difference lies in whether the sequence or arrangement of the selected items is important. In permutation problems like BANANA, different orderings of the letters result in different arrangements.