If you write it like this:

imMunOlogy, 

with one small "m" and one capital "M", and
with one small "o" and one capital "O", 

then you could tell the 4 arrangements:

mOlygoniuM, molygOniuM, MOlygonium, MolygOnium

apart and the answer would be 10!. But that is 4 times too many,
so we divite it by 4.  We divide by 4 because there are 2! ways the o
and O can be arranged time 2! ways the M and the m can be arranged,
so we divide by the factorials of the numbers of times each letter
]s repeated so the answer ie

.

Edwin