Question

Find the number of ways you can arrange (a) all of the letters and (b) 2 of the letters in the given word. FAMILY

Short answer

For part (a), there are 720 ways to arrange all the letters of the word 'FAMILY'. For part (b), there are 30 ways to select and arrange 2 of the 6 letters.

Step-by-step solution

Permutation of all letters

The word 'FAMILY' has 6 distinct letters. Using the formula for permutations, which is \(n!\), where \(n\) is the number of objects, we find that there are \(6! = 720\) ways to arrange all the letters.

Permutation of any two letters

We will apply permutation to the selection of any two letters from 'FAMILY'. The permutation formula we will use is \( nPr = n!/(n-r)! \), where \(n\) is the total number of objects and \(r\) is the number of objects we choose. In this case, \(n\) is 6 (as there are 6 letters in 'FAMILY') and \(r\) is 2 (since we're choosing 2 letters). Using the formula, the calculation is \(6P2 = 6! / (6-2)! = 30.\) Thus, there are 30 ways to select and arrange 2 of the 6 letters.

Key concepts

Factorial Notation

Factorial notation is a mathematical representation for the product of an integer and all the non-negative integers below it. It is denoted by an exclamation point (!). For example, the factorial of 6, written as \(6!\), is calculated by multiplying 6 by each positive integer less than it: \(6 \times 5 \times 4 \times 3 \times 2 \times 1\), which equals 720.

Factorials are fundamental in permutations because they provide a way to calculate the total number of possible arrangements of a set number of objects. Understanding factorial notation is crucial for solving problems that involve permutations, such as arranging letters in a word or selecting a group out of a larger set.

Arrangement of Letters

The arrangement of letters into words or patterns is often explored using permutations, particularly when each letter is unique. Imagine if we wanted to spell a word using individual letter tiles. If we have the word 'FAMILY', the number of ways we can arrange those letters depends on the total number of letters available and whether we are arranging all of them or just a subset.

When arranging all the letters from the word 'FAMILY', each letter can take any position in the arrangement, leading to a multitude of combinations. This concept also extends to situations where only a part of the letters are being arranged, such as selecting 2 letters out of the 6. In both cases, factorial notation plays a pivotal role, transforming an otherwise daunting task into a simple calculation.

Permutation Formula

The permutation formula is a specific mathematical equation used to describe the number of possible arrangements of a subset of items within a larger set. The general form is denoted as \( nPr \) or \( _nP_r \) and is calculated as \( nPr = \frac{n!}{(n-r)!} \), where \(n\) is the total number of items to pick from, and \(r\) is the number of items to arrange.

For example, with the word 'FAMILY', if we want to calculate the number of ways to arrange 2 letters out a total of 6, we use the permutation formula \( 6P2 = \frac{6!}{(6-2)!} \). This equation incorporates the concept of factorial notation to streamline the process. Permutation formulae can get more complex with additional conditions, like if the arrangement must start or end with a certain element or if repetitions are allowed, but the basic formula remains a cornerstone of combinatorial problems.