Question

The number of arrangements that can be made out of the letter of the word " SUCCESS " so that the all S's do not come together is (a) 60 (b) 120 (c) 360 (d) 420

Short answer

The number of arrangements that can be made out of the letters of the word "SUCCESS" so that all S's do not come together is 360.

Step-by-step solution

Calculate the total number of arrangements without restriction

The word "SUCCESS" has 7 letters in total, of which there are 3 S's, 2 C's, and 1 U and 1 E. To find the total number of arrangements, we can use the formula: Total arrangements = \( \frac{n!}{p_1! \cdot p_2! \cdots p_k!}\) where n is the total number of letters, and p are the counts of each repeated letter. In this case, we have: Total arrangements = \( \frac{7!}{3! \cdot 2!}\)

Calculate the number of arrangements where all S's are together

Now, we will treat all the 3 S's as a single unit, and calculate the number of arrangements, considering the other letters around them. We now have 4 "spaces" to arrange: {SSS}, C, C, U, and E. So, to arrange these, we can use the formula just like before: Arrangements with all S's together = \( \frac{5!}{2!}\)

Calculate the difference of Step 1 and Step 2 to find the number of arrangements such that no two S's come together

Now, to find the number of arrangements with no two S's together, we can simply subtract the arrangements found in Step 2 from the total number of arrangements calculated in Step 1. Required arrangements = Total arrangements (Step 1) - Arrangements with all S's together (Step 2) Required arrangements = \( \frac{7!}{3! \cdot 2!} - \frac{5!}{2!}\) Calculating these values, we get: Required arrangements = 420 - 60 = 360 The answer is (c) 360.

Key concepts

Permutations

Permutations are all about arranging items or elements in a specific order. When we talk about permutations, we're interested in knowing how many different ways we can organize a set of things. For instance, in the word "SUCCESS", if we didn't have any repeated letters, we would be looking at finding all possible sequences of these letters. Each unique sequence is called a permutation.
In a permutation:

• Order matters: This means changing the order changes the permutation.
• The arrangement involves the entire set of objects.

Permutations are a fundamental concept in combinatorics because they allow us to count and organize groups of objects in various sequences, especially when the order in which they appear is crucial. Understanding permutations helps not only in solving arrangement problems but also in fields like probability and statistics.

Factorial

A factorial, represented by an exclamation mark (!), is a simple yet powerful mathematical concept used in permutations and combinations. It refers to the product of all positive integers up to a given number "n". For example, 7 factorial, noted as 7!, is:\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]
Factorials are especially useful when calculating the total number of ways to arrange or order a set of items, without any restrictions. When calculating permutations, factorials help in determining the total number of arrangements possible, while adjusting for repeated elements.In our example of the word "SUCCESS", factorials help account for the different letters, especially considering that some of them repeat. Without the factorial concept, permutations involving repeated elements would be harder to grasp.

Repeated Elements

Repeated elements appear when certain objects in a set are indistinguishable from each other. In permutations involving repeated elements, it is essential to adjust calculations to avoid overcounting. For instance, if the set includes items that are identical, like in our word "SUCCESS" where 'S' appears three times and 'C' appears twice, there are fewer unique arrangements than there would be if each item were different.

• Adapt your calculations by dividing the factorial of the total number of items by the factorial of the number of each repeated item.
• This prevents counting identical arrangements multiple times.

The formula used: \[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times ... \times p_k!} \]This formula ensures that we get the correct count of unique permutations by factoring out the overcounted permutations due to indistinguishable elements.

Arrangement Problems

Arrangement problems deal with finding the number of ways we can organize a set of objects according to specific rules or conditions. They are common in questions about permutations, where you might have constraints, like "not all S's can be together" in our "SUCCESS" problem. In these situations, solving the problem typically involves two main steps:

• Calculate the total possible arrangements, without considering any restrictions or special conditions.
• Apply the specific conditions to narrow down the number of acceptable arrangements. This often involves subtracting unwanted arrangements, such as when certain elements must not be together.

Understanding arrangement problems is crucial because they require both basic combinatorial skills and the ability to strategically apply additional constraints or conditions to find the desired solution. This practice enhances logical reasoning skills and prepares you for more complex problems in mathematics.