Question

If all \(S_{4}^{\prime}\) come together, then in how many ways the leters of the word SUCCESSFUL be arranged? (a) 10080 (b) 40080 (c) 2378 (d) none of these

Short answer

a) 180 b) 210 c) 225 d) none of these Answer: d) none of these (240 ways)

Step-by-step solution

Consider the S's and 4's as one unit

We can treat the trio of "S's and 4's" together as a single unit, like this: [SSS4], C, C, U, L. Now we have 5 units (1 block and 4 individual letters).

Calculate the number of permutations of the units

We have 5 units, which can be arranged in 5! ways. However, we have 2 C's, so we need to divide by 2! to account for the repetitions. The total number of permutations is \frac{5!}{2!} = 60.

Calculate the number of permutations of S's and 4's within the block

The block [SSS4] consists of 3 S's and one 4. There are 4! ways to arrange them without considering repetitions. Since there are 3 S's, we need to divide by 3! to account for the repetitions. The total number of permutations within the block is \frac{4!}{3!} = 4.

Calculate the total number of ways to arrange the letters

We have 60 permutations for the 5 units (Step 2) and 4 permutations for the S's and 4's within the block (Step 3). To find the total number of ways, we multiply these two values: 60 * 4 = 240. The answer is 240, which is not one of the given options, so the correct choice is (d) none of these.

Key concepts

Factorial

Factorials are mathematical expressions used to describe the product of an integer and all the integers below it. In other words, the factorial of a number \( n \), symbolized by \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
The concept of factorial is crucial when calculating permutations, as it helps to determine how many different ways an arrangement can be made. Factorials grow very quickly with larger numbers, which is why they are essential for solving various problems involving arrangements and orderings.
When calculating permutations in our exercise, we used factorials to determine the number of ways to arrange 5 units, giving us \( 5! \), and to handle repetitions, we adjust with \( 2! \) for identical Cs.

Combinatorics

Combinatorics is a field of mathematics concerning the counting, arrangement, and combination of objects. It involves studying how objects can be grouped and organized in different ways, often under specific constraints.
This concept becomes handy when dealing with problems related to permutations and combinations, where the order of arrangement may or may not matter. In our exercise, we specifically dealt with permutations, which meant we cared deeply about the order of the letters.
Understanding that combinatorics can often involve restrictions, such as repeated elements, is fundamental as it allows us to adapt our counting strategies to accurately represent the problem task.

Letter arrangement

Letter arrangement in combinatorics involves determining how letters can be arranged in different sequences. Here, each arrangement constitutes a permutation of the letters.
In the given exercise, the challenge of arranging letters from the word "SUCCESSFUL" requires accounting for both the placement and the repetitions among them.
The approach starts with treating blocks of letters as single units, particularly because some letters are considered to be grouped together, like the triplet of S's and the number 4. This enables a simpler computation, as it reduces the complexity down to manageable units that we can handle with factorial mathematics.

Repetition handling

Handling repetitions is a crucial step in permutation problems as repeated elements influence the total number of possible arrangements.
When identical items are present, factoring this into our permutation calculations ensures we don't overestimate the number of unique arrangements.
In our exercise, repetitions are found in the form of two C's and three S's. To adjust for this, we divide our permutation by the factorial of the number of repeated items, eg. using \( 2! \) for the Cs and \( 3! \) for the Ss.
This method ensures accurate results when calculating possible arrangements, as it properly accounts for indistinguishable permutations of identical items.