Question

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

Short answer

For part (a), there are 6 different ways to arrange all the letters in 'TRY'. For part (b), there are 3 different ways to select any 2 letters from 'TRY'

Step-by-step solution

Find Total Permutations for all letters

A permutation is the number of ways you can arrange a set of items. Since there are 3 distinct letters in the word 'TRY', using the formula for permutations \( n! \), where \( n \) is the number of items and ‘!’ indicates factorial, we get \( 3!= 3*2*1 = 6 \) Hence, there are 6 different ways to arrange all the letters in 'TRY'.

Find Combinations for 2 letters

A combination is the number of ways you can select items from a larger set. So, to find the number of ways to select any 2 letters from 'TRY', we need to use the formula for combinations \( \binom{n}{r} \), where \( n \) is the total number of items and \( r \) is the number of items we are selecting. Hence, \( \binom{3}{2} = 3 \). Thus, there are 3 different ways to select 2 letters from 'TRY'.

Key concepts

Factorial Notation

Factorial Notation is a mathematical concept used to describe the product of an integer and all the integers below it, down to 1. It’s denoted by an exclamation mark (!). For instance, the factorial of 4 (written as 4!) is calculated as 4 × 3 × 2 × 1, which equals 24. One of the key uses of factorial notation is in counting permutations, where we want to learn in how many ways we can arrange different objects.

It's important to remember that the factorial of zero (0!) is always 1. This is a convention in mathematics, making some formulas easier to work with, as it aligns with the idea that there is exactly one way to arrange zero objects.

In the context of the given problem, to arrange all 3 letters in 'TRY', we calculate 3!, which amounts to 3 × 2 × 1 = 6. This implies there are 6 distinct ways to arrange the letters 'T', 'R', and 'Y.'

Arranging Letters

Arranging letters of a word can be a common exercise to understand permutations. The permutations of a set of elements are all possible arrangements of those elements. When the elements are letters, each different arrangement can form a new word or a variation of the original, even if it doesn't make sense linguistically.

For the given exercise with the word 'TRY', we consider each letter unique and use factorial notation to calculate the total permutations. For three unique letters, there are 3! (factorial of 3) ways to arrange them, resulting in 6 different permutations, each representing a possible arrangement of those three letters.

Mathematical Combinations

Mathematical combinations allow us to count the number of ways a certain number of items can be selected from a larger group, where the order of selection doesn't matter. Unlike permutations, combinations do not consider different orders as distinct.

To calculate combinations, we use the formula \( \binom{n}{r} \), also called 'n choose r'. Here, \( n \) represents the total number of items, while \( r \) is the number of items to select. For our exercise, selecting 2 letters from 'TRY', we're looking at \( \binom{3}{2} \) combinations. This calculation decides that there are 3 possible ways to select 2 letters out of 3, disregarding the order of selection, leading to our answer of 3.