Use an appropriate permutations formula to solve.

How many arrangements can be made using four of the letters of the word NUMBER if no letter is to be used more than once?

Arrangements
(Type a whole number.)

To solve the question of how many arrangements can be made using four of the letters of the word 'NUMBER' without repeating any letter, we will use the formula for permutations.
The word 'NUMBER' consists of 6 distinct letters. When choosing any 4 letters from these 6, we are interested in the number of permutations (or arrangements) of these 4 letters. This scenario is a permutation problem because the order in which we arrange these letters matters.
The formula for permutations, where we are arranging r items out of a total of n items, is given by:
P ( n , r ) = ( n − r )! n ! ​
In this problem:

n = 6 (the total number of letters in 'NUMBER')
r = 4 (the number of letters to arrange)

Plugging these values into the permutation formula:
P ( 6 , 4 ) = ( 6 − 4 )! 6 ! ​ = 2 ! 6 ! ​
Calculating the factorials:

6 ! = 6 × 5 × 4 × 3 × 2 × 1 = 720
2 ! = 2 × 1 = 2

Thus, the number of permutations is:
P ( 6 , 4 ) = 2 720 ​ = 360
Therefore, there are 360 different arrangements that can be made using four of the letters of the word 'NUMBER' without repeating any letter.

Answered by LucasMatthewHarris | 2025-07-21

The total number of arrangements that can be made using four of the letters from the word 'NUMBER' is 360. This is calculated using the permutations formula P ( n , r ) , where n = 6 and r = 4 . By applying the formula, we find the answer to be 360 arrangements.
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Answered by LucasMatthewHarris | 2025-08-06