Question

Using all the letters of the word 'QUESTION', how many different words can be formed? (1) \(8 !\) (2) \(7 !\) (3) \(7 \times 7 !\) (4) \(9 !\)

Short answer

Answer: (1) \(8!\)

Step-by-step solution

Identify the number of distinct letters

In the word 'QUESTION', there are 8 distinct letters: Q, U, E, S, T, I, O, and N.

Calculate the permutations

Since there are 8 distinct letters, we need to calculate the permutations for these 8 letters. The formula for permutations is the factorial of the number of elements. Here, there are 8 elements, so the number of permutations is 8! (8 factorial). 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320 So, there are 40,320 different words that can be formed using all the letters of the word 'QUESTION'. The correct answer is (1) \(8!\).

Key concepts

Factorial

When we talk about the concept of a factorial, denoted by the exclamation mark (!), we refer to a fundamental principle in mathematics used to calculate the total number of ways to arrange a set of objects. The factorial function is simple: the factorial of a positive integer n, written as n!, is the product of all positive integers less than or equal to n. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Understanding factorials is essential since they form the basis of permutations, one of the key concepts in combinatorics. To improve comprehension, it's helpful to visualize factorial as the number of ways to arrange books on a shelf or seats in a row, which distinctly conveys the idea of sequential arrangement without repetition.

Permutations and Combinations

Permutations and combinations are two related, yet different concepts in the study of combinatorics. While both deal with the arrangement of items, they differ based on whether the order of arrangement matters or not.

• Permutations refer to the number of ways a set of objects can be arranged in order where the sequence is important. Using the factorial function, we calculate permutations to determine the linear arrangements of distinct items. In the given problem, permutations were used to find out the different arrangements of the letters in the word 'QUESTION'.
• Combinations, on the other hand, are about groupings where the order isn't important. This concept is typically applied when we need to figure out how many different groups or selections can be made from a larger set.

To firmly grasp these concepts, imagine permutations as the different ways runners can finish a race (first, second, third, etc.), while combinations are like choosing two fruits from a basket regardless of the order they are picked.

Combinatorics

Combinatorics is the branch of mathematics that studies counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is foundational in fields such as probability, algebra, and geometry. Deeply connected to the concepts of permutations and combinations, combinatorics explores the various ways of arranging and selecting items within a finite set.

To further develop an intuitive understanding of combinatorics, think about solving puzzles that require arranging or selecting pieces with certain constraints, such as a jigsaw puzzle (permutations) or a card hand from a deck (combinations). Real-world applications span multiple areas, from determining possible genetic variations in biology to optimizing computer network configurations in information technology. Students can enhance their understanding by practicing different combinatorial problems and by visualizing these scenarios through real-life examples.