How many five-letter words can be made from the English alphabet with no repeating letters
The correct answer and explanation is:
Correct Answer: 7,893,600
To find the total number of five-letter words that can be formed from the English alphabet with no repeating letters, start by considering how many choices there are for each letter. The English alphabet has 26 letters. For the first letter of the word, there are 26 possible choices. Once one letter has been used, it cannot be used again, leaving 25 choices for the second letter. For the third letter, there will be 24 remaining choices, then 23 choices for the fourth letter, and finally 22 choices for the last letter.
This situation is an example of a permutation since the order in which the letters are arranged matters. The general formula for the number of permutations of kk items from nn distinct items is given by P(n,k)=n!/(n−k)!P(n, k) = n! / (n – k)!. In this problem, n=26n = 26 and k=5k = 5. So the number of possible five-letter arrangements is:
P(26,5)=26×25×24×23×22P(26, 5) = 26 \times 25 \times 24 \times 23 \times 22.
Calculating this step by step:
26 times 25 equals 650.
650 times 24 equals 15,600.
15,600 times 23 equals 358,800.
358,800 times 22 equals 7,893,600.
Therefore, there are exactly 7,893,600 possible five-letter arrangements without any repeated letters.
This calculation assumes that every sequence of five distinct letters counts as a “word,” whether or not it is meaningful in English. This is common in combinatorics problems, where “word” just means an ordered sequence. If the problem asked for only actual English words, a dictionary or word list would be needed, and the total would be far smaller. But when counting possible arrangements, any combination that meets the conditions is valid.