<p>How many words can be formed using all the letters of the word <strong>ASSASSINATION</strong>?</p>
Step-by-Step Solution
Key Concept: Count the frequency of each repeated letter, then apply the formula for permutations with repetition: n!/(n₁! × n₂! × ... × nₖ!), where n is total letters and nᵢ are frequencies of identical letters.
<p><strong>Step 1:</strong> Count total letters in ASSASSINATION: A-S-S-A-S-S-I-N-A-T-I-O-N = 13 letters</p><p><strong>Step 2:</strong> Count frequency of each letter:</p><ul><li>A appears 3 times</li><li>S appears 4 times</li><li>I appears 2 times</li><li>N appears 2 times</li><li>T appears 1 time</li><li>O appears 1 time</li></ul><p><strong>Step 3:</strong> Apply the formula for permutations with repetition:</p><p>Number of words = \(\dfrac{13!}{3! \times 4! \times 2! \times 2! \times 1! \times 1!}\)</p><p>Since 1! = 1, we can omit them:</p><p>∴ Answer: \(\dfrac{13!}{3! \times 4! \times 2! \times 2!}\)</p>
Correct Answer: \(\dfrac{13!}{3! \times 4! \times 2! \times 2!}\)