Q:

combinatorics counting words

How many different 5-letter words (= strings) can be formed from the
letters in CALCULUS if no letter can be used more times than it occurs in
the word CALCULUS? For example, CCUUL and LSUCS would count but
UUCSU would not count.

A:

you have $CC,A,LL,UU,S$
Possible cases are:
$1.)$Two identical,Two identical,One distinct
$2.)$Two identical,Three distinct
$3.)$All distinct
EDIT:
For case $1.)$ ways of selection = ${3\choose2}{3\choose 1}$ and thus total such words= ${3\choose2}{3\choose 1}\frac{5!}{2!2!}$
For case $2.)$ ways of selection = ${3\choose1}{4\choose 3}$ and thus total such words= ${3\choose1}{4\choose 3}\frac{5!}{2!}$
For case $3.)$ ways of selection = ${5\choose5}$ and thus total such words= ${5\choose5}5!$

