Solution: First, compute the total number of distinct arrangements of "BANANA" without restrictions. The word has 6 letters with repetitions: 3 A's, 2 N's, and 1 B.
["Solution: Computing Total Arrangements of "BANANA" Without Restrictions", "When solving permutation problems involving repeated letters, it’s essential to adjust the general formula to avoid overcounting. The word “BANANA” provides a classic example: it contains 6 letters with repetitions — 3 A’s, 2 N’s, and 1 B. In this article, we’ll walk through the step-by-step solution to compute the total number of distinct arrangements of “BANANA” without restrictions, leveraging combinatorial principles.", "---", "### Step 1: Recall the General Formula for Permutations with Repetitions", "For a word with total letters ( n ), containing repeated letters with frequencies as follows:\n- Letter X appears ( f_X ) times\n- Letter Y appears ( f_Y ) times\n- …\n- Letter Z appears ( f_Z ) times", "the total number of distinct permutations is given by:", "[\n\ ext{Total Arrangements} = \frac{n!}{f_X! \ imes f_Y! \ imes \cdots \ imes f_Z!}\n]", "This formula accounts for the fact that rearranging identical letters among themselves produces indistinguishable arrangements.", "---", "### Step 2: Identify Parameters for “BANANA”", "From “BANANA”:\n- Total letters, ( n = 6 )\n- B occurs 1 time\n- A occurs 3 times\n- N occurs 2 times", "---", "### Step 3: Apply the Formula", "Plug values into the formula:", "[\n\ ext{Distinct Arrangements} = \frac{6!}{3! \ imes 2! \ imes 1!}\n]", "Calculate each factorial:\n- ( 6! = 720 )\n- ( 3! = 6 )\n- ( 2! = 2 )\n- ( 1! = 1 )", "Now compute:", "[\n\frac{720}{6 \ imes 2 \ imes 1} = \frac{720}{12} = 60\n]", "---", "### Step 4: Interpret the Result", "The total number of distinct arrangements of the letters in “BANANA” is 60. This calculation respects the repetitions of A and N, ensuring each unique permutation is counted only once without overcounting due to identical letters.", "---", "### Why This Matters", "Understanding how to count distinct permutations with repeated characters is crucial in combinatorics, real-world applications, and algorithm design — especially in fields like linguistics, cryptography, and data organization where pattern uniqueness impacts outcomes.", "---", "### Key Takeaway", "To find distinct arrangements of words with repeated letters:\n1. Count total letters.\n2. Identify and count repetitions of each character.\n3. Apply the permutation-with-repetitions formula: divide the factorial of total letters by the product of factorials of each repetition count.\n4. The result is the number of unique, unordered arrangements.", "---", "Conclusion:\nComputing the total number of distinct arrangements of “BANANA” yields 60 unique permutations, properly adjusting for repeated letters through factorial division. This method delivers accurate, efficient results for identical-letter word arrangements.", "---", "Keywords: permutations of BANANA, distinct arrangements, total arrangements without restrictions, repeated letters formula, combinatorics, factorial division, BANANA permutations, distinct word arrangements."]
