Problem 159 - A Complete Resource Book in Mathematics for JEE Main

Question

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two \(\mathrm{S}\) are adjacent? (A) \(8 \cdot{ }^{6} C_{4} \cdot{ }^{7} C_{4}\) (B) \(6 \cdot 7 \cdot{ }^{8} C_{4}\) (C) \(6 \cdot 8 \cdot{ }^{7} C_{4}\) (D) \(7 \cdot{ }^{6} C_{4} \cdot{ }^{8} C_{4}\)

Step-by-Step Solution

Answer

The answer is option (C): \(6 \cdot 8 \cdot \binom{7}{4}\).

Step 1: Count Total Letters

The word MISSISSIPPI has 11 letters composed of 1 M, 4 I's, 4 S's, and 2 P's.

Step 2: Select Slots for Non-S Letters

Consider the letters M, I, and P first. Arranging these letters means placing the 1 M, 4 I's, and 2 P's, which totals 7 letters. There are \(7!\) ways to arrange these letters.

Step 3: Determine Positions for S

With the 7 arranged letters of M, I, and P, there are 8 gaps created (before, between, and after letters) where we need to place the S's such that no two S's are adjacent.

Step 4: Choose Positions for S

To ensure the S's are not adjacent, choose 4 gaps from the 8 available. This can be done in \(\binom{8}{4}\) ways.

Step 5: Calculate Total Arrangements

For each gap configuration of selecting S's, arrange the remaining 7 letters (M's, I's, and P's). Each configuration can be arranged in \(\frac{7!}{4!2!1!}\) ways since we must account for identical I's and P's.

Step 6: Compute Final Answer

Combine the ways to select S positions and the arrangements of the other letters into the formula \(\binom{8}{4} \times \frac{7!}{4!2!}\). This value coincides with option (C) \(6 \cdot 8 \cdot \binom{7}{4}\).

Key Concepts

Permutations

Permutations are an essential concept in combinatorics. They refer to different arrangements or orders of a set of items. When dealing with permutations, every item must be used, and the arrangement is important. For example, the word "MISSISSIPPI" consists of letters that can be rearranged to form different sequences. When calculating permutations, if items within the set are unique, you can use the formula for the total number of permutations, which is given by the factorial of the number of items, denoted as \( n! \).

• The factorial, \(m!\), of a number \(m\) is the product of all positive integers up to \(m\).
• For the word "MISSISSIPPI", there are 11 letters, so the initial calculation for permutations would be \(11!\).

However, since some letters repeat, such as the I's and S's, we must adjust the permutation formula. We divide by the factorial of each group of identical items. Hence, the correct expression becomes \( \frac{11!}{4!4!2!} \). This accounts for 4 I's, 4 S's, and 2 P's. By using this method, you ensure that identical letters are not counted multiple times in different spots.

Combinations

Combinations are different from permutations in that the order of the items does not matter. In our example with the word "MISSISSIPPI", if we had to choose positions for certain letters like S's without worrying about their order, we would use combinations.Combinations are calculated using the formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), which represents "n choose r", or the number of ways to choose \(r\) items from \(n\) total items, disregarding order. In the step-by-step solution for "MISSISSIPPI", we used combinations to choose slots for the S's. After arranging the other letters, there were 8 gaps. We selected 4 out of these 8 gaps to place the S's using combinations, specifically \( \binom{8}{4} \). This ensures that no two S's are adjacent, as the choice of gaps determines their separation.Using combinations is crucial when placing items into positions where the order specifically doesn't matter, such as positioning letter groups in a sequence without regard to the actual order of those positions.

Arrangement of Letters

Arranging the letters in a word involves understanding both permutations and combinations. In the "MISSISSIPPI" exercise, we applied these concepts to form distinct words under certain conditions.First, we arranged the non-S letters: M, I's, and P's, taking into account identical letters. This required a permutation calculation with adjusted factorials: \( \frac{7!}{4!2!1!} \) for 1 M, 4 I's, and 2 P's.

• We consider each unique letter and the repetition of other letters.
• The gaps between the arranged letters then create possible positions for S's.

The key to solving the exercise was ensuring no two S's were adjacent. We navigated this by identifying the 8 gaps between and around the 7 arranged letters (M, I's, and P's). We selected gaps using combinations, ensuring a correct spread of S’s.In summary, to arrange letters like in "MISSISSIPPI", the process involves:

• Calculating permutations of unique letters adjusted for repetitions.
• Using combinations to place certain letters under restrictions (like non-adjacency).

A combination of these approaches provides a clear path to finding all valid arrangements.