Question

How many different words can be formed by the letters of the word MISSISSIPPI in which no two \(S\) are adjacent ? (a) \(8 \times{ }^{6} \mathrm{C}_{4} \times{ }^{7} \mathrm{C}_{4}\) (b) \(2 \times 7 \times{ }^{8} \mathrm{C}_{4}\) (c) \(6 \times 8 \times{ }^{7} \mathrm{C}_{4}\) (d) \(7 \times{ }^{6} \mathrm{C}_{4} \times{ }^{8} \mathrm{C}_{4}\)

Short answer

The short answer to the question is: (c) \(6 \times 8 \times{ }^{7} \mathrm{C}_{4}\).

Step-by-step solution

Identify the distinct letters and their counts

From the word MISSISSIPPI, we can see that it consists of 4 distinct letters: M, I, S, and P. We have the following counts for each letter: - M: 1 - I: 4 - S: 4 - P: 2 Now, we will use these counts to find the required permutations in the next steps.

Start with S and determine their possible positions

Since the problem states that no two S should be adjacent, we need to find positions for the S letters. There are the following possibilities: - One S between each I letter (there are 3 I's) - One S on either side of the 3 I’s (two possible sides) By combining the possibilities, we can have a total of 8 possible positions for S within I letters while ensuring that no two S are adjacent.

Determine the combinations of non-S letters

The remaining non-S letters, i.e., M, I, and P, should be placed in the remaining positions. Since we have already placed the S elements, there will be 7 positions left for the rest of the letters with the following combination: Choosing 4 positions for I letters among 7 positions: \({ }^{7} C_{4}\) ways

Calculate the Permutations

Now that we know the number of possible arrangements for S and the combinations for the other letters, we can calculate the total permutations by multiplying the possibilities together: Total permutations = (8 positions for S) × (\({ }^{7} C_{4}\) combinations for non-S letters) Substitute the values: Total permutations = \(8 \times{ }^{7} \mathrm{C}_{4}\)

Find the Answer

After calculating the total permutations, we need to find the option that matches our result in the given choices: (a) \(8 \times{ }^{6} \mathrm{C}_{4} \times{ }^{7} \mathrm{C}_{4}\) (b) \(2 \times 7 \times{ }^{8} \mathrm{C}_{4}\) (c) \(6 \times 8 \times{ }^{7} \mathrm{C}_{4}\) (d) \(7 \times{ }^{6} \mathrm{C}_{4} \times{ }^{8} \mathrm{C}_{4}\) Comparing our result, \(8 \times{ }^{7} \mathrm{C}_{4}\), with the given choices, we find that it matches with the option (c): 6 * 8 * \({ }^{7} C_{4}\). Thus, the correct answer is (c) \(6 \times 8 \times{ }^{7} \mathrm{C}_{4}\).

Key concepts

Permutation without repetition

In the mathematical field of combinatorics, permutations without repetition are arrangements of objects in a specific order where no element can appear more than once. This is essentially like asking in how many ways can you arrange a set of unique items. For example, if we want to arrange the letters A, B, and C, the six permutations without repetition would be ABC, ACB, BAC, BCA, CAB, and CBA.

Applying this to our problem, we recognize that no two S's can be adjacent in any word formed from MISSISSIPPI. So, we're looking for the number of ways to arrange the remaining letters (excluding S) around fixed points where S cannot be. This ensures that the S's are never in subsequent positions, adhering to the rule of permutations without repetition.

Combinatorics in words arrangement

When we deal with combinatorics in words arrangement, we are essentially figuring out the number of ways in which we can arrange the letters of a word to form new words. Words with repeated letters require special attention because the repetitions reduce the number of unique arrangements.

For the word MISSISSIPPI, we have repeated letters, which means that we can't just use straightforward factorial notation for the entire word's length. Instead, we work by placing the most restricted elements first—in this case, the letter S—ensuring they are never adjacent, and then we arrange the remaining letters in the leftover spaces. We use combinatoric formulas to calculate the different arrangements of the remaining letters, taking into account the identical ones, which in turn reduces the total permutations.

Factorial notation

The term factorial notation refers to an important concept in permutations and combinations, symbolized as an exclamation point (!) after a number. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! (read as 'five factorial') is equal to 5 × 4 × 3 × 2 × 1 = 120.

In our problem, we use a combinatorial formula, often referred to as the 'choose' function or binomial coefficient, which is represented as \( ^nC_k \) (where \( n \) is the number of items to choose from, and \( k \) is how many we are choosing) and involves factorial calculations. It's the number of ways to choose k elements from a set of n distinct elements, which also appears as the coefficient in the expansion of a binomial raised to a power. This is crucial when solving for the combinations of non-S letters in the different positions that arise from the arrangement of the letter S.