Question

The number of 5 letter words formed using letters of word "CALCULUS" is

• A. $280$
• B. $15$
• C. $1110$
• D. $56$

Answer 1

The correct option is C $1110$

The letters are CC LL UU AS

Taking $5$ different words

No. of ways$=\underset{All5selected}{{}^5{C}_5}\times \underset{arrangement}{5!}=120$

Eg CLUAS

$2$ same, $3$ different $=\underset{1pair}{{}^3{C}_1}\times \underset{3different}{{}^4{C}_3}\times \frac{5!}{2!}=720$

Eg CC LUA

$2$ same $2$ same $1$ different$=\underset{2pair}{{}^3{C}_2}\times \underset{1different}{{}^3{C}_1}\times \frac{5!}{2!2!}=270$

Eg CC UU L

or CC UU A

$3$ same is not possible

i.e., $3$ or more same letters are not present.

$\Rightarrow$ Total words$=120+720+270$

$=1110$.