Question

$10$ different letters of English alphabet are given. Out of these letters, words of $5$ letters are formed. How many words are formed when at least one letter is repeated?

• A. $69760$
• B. $98748$
• C. $96747$
• D. $97147$

Answer 1

The correct option is A

$69760$

Explanation for the correct option:

Step 1. Find the number of words formed:

As we know,

Words which have at least one letter repeated $=$ all words possible - words with no letter repeated

Step 2. Find all words possible:

Such word can have $10$ alphabets as first letter, 10 letters for second letter and so on.

$\Rightarrow$all words possible $=10\times 10\times 10\times 10\times 10$

$=100000$

$={10}^5$

Step 3. Find words with no letter repeated:

Such word can have $10$ alphabets as first letter, 9 letters as second letter and so on

$\Rightarrow$words with no letter repeated $=10\times 9\times 8\times 7\times 6$

$=30240$

$={}^{10}P_5$

$\therefore$Words which have at least one letter repeated

$\Rightarrow {10}^5-{}^{10}P_5=100000-30240$

$=69760$

Hence, Option ‘A’ is Correct.