Question

With $ 10 $ different letters, $ 5 $ letter words are formed. Then the number of words which have at least one letter repeated is

Answer 1

Hint: Here words which have at least one letter repeated can be found by taking the difference between all the possible words and the words with no letter repeated. No letters repeated can be calculated by taking the concepts of factorials.

Complete step by step solution:
First of all we will find the total words possible in such a way that ten alphabets as first letter, ten letters for second letter and similarly so on for five lettered words.
It gives all possible words $ = 10 \times 10 \times 10 \times 10 \times 10 $
Simplify the above expression finding the product of the above expression –
Total possible words $ = 100000 $ ….. (A)
Now, to get the words with no letter repeated can have ten alphabets as first letter, nine letters as second letter and similarly so on for five lettered words.
So, the total words with no letter repeated $ = 10 \times 9 \times 8 \times 7 \times 6 $
Find the product of the terms in the above expression and simplify.
Total words with no letter repeated $ = 30240 $ …… (B)
Now, by using the values of the equation (A) and (B)
Thus, words having at least one letter repeated $ = 100000 - 30240 $
Find the difference of the terms in the above expression –
Words having at least one letter repeated $ = 69760 $
This is the required solution.
So, the correct answer is “69760”.

Note: Always read the word statement properly. Here we have calculated using the difference of no repetition of the letters, since it was given at least one so it can be more than one and therefore it can be one time, two, three, four and five times repetition of the letters and it can be solved by adding these numbers but it may create the lengthy solutions.