Question

Using the letters of the word PRIVATE, how many 6-letter words can be formed which begin with \(\mathrm{P}\) and end with E? (1) \(3 !\) (2) \(4 !\) (3) \(7 !\) (4) \(5 !\)

Short answer

Answer: (2) \(4 !\) (120 different 6-letter words can be formed)

Step-by-step solution

Identify the number of available spots and valid letter options

We know that the 6-letter word starts with "P" and ends with "E". So, there are 4 spots left for permutations, and there are 5 remaining letters (R, I, V, A, T) to fill those 4 spots.

Calculate the number of permutations

The number of permutations for 4 spots with 5 possible unique letters can be calculated using a factorial. To do this, we multiply the number of possible choices in each spot: \(5!/(5-4)! = 5 \times 4 \times 3 \times 2 \times 1 / 1 = 1 \times 2 \times 3 \times 4 \times 5 = 120\)

Choose the correct answer from the options

We found that there are 120 different 6-letter words that can be formed using the letters of the word "PRIVATE", which begin with P and end with E. So the correct answer is: (2) \(4 !\)

Key concepts

Factorial Notation

Factorial notation is fundamental in the field of mathematics, particularly when dealing with permutations and combinations. A factorial, represented by an exclamation point (!), is the product of all positive integers up to a given number. For instance, the factorial of 5, written as \(5!\), is calculated by multiplying 5 by every positive integer less than it: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

The factorial value grows extremely fast, making it a crucial component for solving problems related to the arrangement or selection of objects. Factorials are the building blocks of combinatorics, the branch of mathematics dealing with counting combinations and permutations. Understanding this notation is essential for solving the exercise as it directly relates to the ways letters can be arranged in the word 'PRIVATE'.

Combinatorics

Combinatorics is a vast area of mathematics that focuses on counting, arranging, and combining sets in specific ways. It includes studies of combinations, permutations, and other mathematical structures. Combinatorics helps in solving problems about constructing subsets, designing experiments, and even developing various kinds of computer algorithms.

One practical application of combinatorics is determining the number of ways to arrange objects, which we often encounter when dealing with puzzles, games, or in this case, word formations from a set of letters. The discipline employs many principles and formulas, such as the rule of sum and rule of product, which inform our approach to the provided exercise on creating 6-letter words from the letters of 'PRIVATE'.

Permutation and Combination

Permutation and combination are methods used to count the arrangements within a set when the order does or doesn't matter, respectively. Permutation is concerned with the arrangement of all or part of a set of objects, with attention to the order. For instance, the arrangement of letters to form words where 'PVT' is different from 'VTP'. This is where factorial notation comes in handy, as it is used to calculate the number of possible permutations.

Combination, on the other hand, focuses on the selection of objects where the order is irrelevant, such as the combination of ingredients in a salad. In our exercise with the word 'PRIVATE', permutation is used to solve the problem since the arrangement of letters is essential. To clarify, we use \(5!/(5-4)!\) to find the number of ways to arrange 5 unique letters in 4 spots, leading to the correct option, which reflects the number of permutations possible.