Question 42 of 42intermediate🔧 ApplyNumerical3 marks

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?

Correct Answer

151200

Exercise: Miscellaneous Exercise on Chapter 6 | Q: 11 | (Page 24)
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Explanation

This question follows the same approach as Example 14 in the textbook, where vowels are treated as a single object when they must occur together. Here, all 4 S's must be together, so we treat them as one single object (SSSS). The word ASSASSINATION has 13 letters total: A appears 3 times, S appears 4 times, I appears 2 times, N appears 2 times, T appears once, and O appears once. When S's are grouped as one object, we effectively have 10 objects to arrange. Using the formula for permutations with repetition from the context, we divide by factorials of repeated letters.

Solution Steps

  1. Step 1: Count the letters in ASSASSINATION. Total = 13 letters. A appears 3 times, S appears 4 times, I appears 2 times, N appears 2 times, T appears 1 time, O appears 1 time.

  2. Step 2: Treat all 4 S's as one single object (SSSS). Now we have 10 objects to arrange: (SSSS), A, A, A, I, I, N, N, T, O.

  3. Step 3: Apply the formula for permutations with repetition: Number of arrangements = 10! / (3! × 2! × 2!) where 3! is for A's, 2! is for I's, and 2! is for N's.

  4. Step 4: Calculate: 10! / (3! × 2! × 2!) = 3628800 / (6×26 \times 2 × 2) = 3628800 / 24 = 151200.

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