Question

Ten different letters of the alphabet are given. Words with five letters are formed from these given letters. Then the number of words that have at least one letter repeated are:

• A. $69760$
• B. $30240$
• C. $99748$
• D. None of these

Answer 1

The correct option is A

$69760$

Explanation for correct answer:

Total number of words that can be formed are ${10}^5=100000$

Total number of words formed without repetition are ${}^{10}{P}_5$

now to calculate ${}^{10}{P}_5$we use the formula ${}^n{P}_n=1\times 2\times ...........\times n$ so we get

$$\begin{gathered} {}^{10}{P}_5=10\times 9\times 8\times 7\times 6 \\ =30240 \end{gathered}$$

the number of words that have at least one letter repeated is calculated by $=$total number of words formed $-$number of words formed without repetition

$$\begin{gathered} \Rightarrow 100000-30240 \\ \Rightarrow 69760 \end{gathered}$$

Hence, the correct answer is option (A)