Question

Ten different letters of the English alphabet are given. Out of these a word is formed using $5$ letters with replacement. The probability that atleast one letter is repeated in the word is

• A. $\frac{1}{{10}^5}$
• B. $1-\frac{{}^{10}{P}_5}{{10}^5}$
• C. $1-\frac{{}^{10}{P}_5}{10!}$
• D. $\frac{{}^{10}{P}_5}{{10}^5}$

Answer 1

Answer: (B)

$1-\frac{{}^{10}{P}_5}{{10}^5}$

Without restriction
Number of arrangements of $5$ letter out of $10$ different letters $={10}^5$
To find 'atleast one', it is better to find the probability of the complement of the event (i.e. no letter is repeated).
$\therefore$ Number of arrangement of $5$ letter $($out of $10$) without repetition ${=}^{10}{P}_5$
$\Rightarrow P($ no letter is repeated $)=\frac{{}^{10}{P}_5}{{10}^5}$
$\Rightarrow P($ atleast one letter is repeated $)=1-\frac{{}^{10}{P}_5}{{10}^5}$
Hence the required probability $1-\frac{{}^{10}{P}_5}{{10}^5}$