$\text{The number of five-letter words formed with the letters of the word CALCULUS is }$

C-2,   A-1,  L-2,  U-2,  S-1

This can be done in 3 ways

2identical+2identical+1 different  =    $\frac{5!}{2! 2!1!} 3_{c_2 } 3_{c_1} = 270 (3_{c_{2 }}to select 2 from C,L,U, 3_{c_{1 }}to select 1 from remaining three)$   

$$\begin{gathered} \text{2 identical+3 different = }\frac{5!}{2!} 3_{c_1} 4_{c_{3 }} \\ = 720 \end{gathered}$$

$( 3_{c_{1 }}to select 1 repeating letter , 4_{c_3}to select from remaining 2 repeated and 2 non repeated letters)$

$$\begin{gathered} All 5 different letters =5!=120 \\ total number of words= 1110 \end{gathered}$$