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. $30240$
• B. $98748$
• C. $96747$
• D. $97147$

Answer 1

The correct option is A $30240$

Of the ten letters $5$ letter can be selected in ${{}^{10}C}_5$ ways.

Their arrangements are $5!{{}^{10}C}_5$

In these no letter is repeated.

In all other at least use letter is repeated.

Total choices are $10!\left({{}^{10}C}_{10}\right)$

Therefore answer $={{}^{10}P}_5-{{}^{5}P}_5=\frac{10!}{(10-5)!}-\frac{5!}{(5-5)!}=\frac{10!}{5!}=\frac{\mathrm{10.9.8.7.6.5}!}{5!}=540\times 50=30240$