How Many Ways Can 4 Letters of the Word 'Number' Be Arranged?

The word 'number' is a six-letter word with all distinct letters. Arranging specific combinations or all four letters of 'number' without repetition is a common permutation problem in combinatorial mathematics. This guide explains different ways to solve this problem and explores the underlying principles of permutations.

Calculating Permutations of 4 Letters from 'Number'

The word 'number' contains 6 different letters. When arranging 4 out of these 6 letters without repetition, the total number of arrangements can be calculated using the formula for permutations, (nPr). The formula is given by:

( P(n, r) frac{n!}{(n-r)!} )

In this case, (n 6) and (r 4), so the permutation formula becomes:

( P(6, 4) frac{6!}{(6-4)!} frac{6!}{2!} )

To calculate, we need to evaluate the factorials:

( 6! 6 times 5 times 4 times 3 times 2 times 1 720 )

( 2! 2 times 1 2 )

Substituting these values into the permutation formula gives:

( P(6, 4) frac{720}{2} 360 )

Thus, there are 360 different ways to arrange 4 letters from the word 'number' without repetition.

Alternative Methods to Calculate Permutations

Additionally, the permutations can be calculated in multiple equivalent ways. Here are two alternative approaches to verify that the result is 360:

Step-by-Step Calculation

One approach is to use the step-by-step calculation method:

1. Choose 4 letters out of 6 in (binom{6}{4}) ways. This can be calculated as:

( binom{6}{4} frac{6!}{4!(6-4)!} frac{720}{24 times 2} 15 )

2. For each combination of 4 letters, there are (4!) ways to arrange them. This is calculated as:

( 4! 4 times 3 times 2 times 1 24 )

3. Therefore, the total number of permutations is:

( 15 times 24 360 )

This confirms that there are 360 distinct ways to arrange 4 letters from 'number'.

Combining Selection and Arrangement

An alternative method is to directly combine the selection and the arrangement steps:

1. First, choose 4 letters from the 6 available letters. This can be done in (binom{6}{4}) ways, which again equals 15.

2. For each chosen set of 4 letters, there are (4!) ways to arrange them, which equals 24.

Hence, the total number of permutations is:

( 15 times 24 360 )

This method provides an alternative but equivalent way to calculate the total permutations.

Understanding permutations is crucial in various fields, including computer science, cryptography, and combinatorial mathematics. Whether using the direct formula or step-by-step methods, the result remains consistent and accurately reflects the number of possible arrangements of 4 letters from the word 'number'.