SOLUTION: If all the distinguishable arrangements of the word "tutor" were listed in alphabetical order and numbered #1,#2,#3,..., what would be the number of the arrangement "tutor"?

There are  distinguishable arrangements of "tutor".

The letters in "tutor" in alphabetical order are "o","r","t","t",u". So
arrangement #1 would be "orttu" and #60 would be "uttro".

Therefore "tutor" would be much closer to the bottom of the list than
to the top.  So it will be easier to start at the bottom of the list
and go upward to find "tutor" than to start at the top and go downward 
to it.  

The last arrangements in the list begin with "u".

Now we find out the number of  arrangements at the bottom of the list begin with
"u".  In each of those, the "u" is followed by a distinct arrangement of the
letters "o","r","t","t", of which there are . So the last 12
arrangements begin with "u", so these 12 arrangements:

#60,#59,#58,$57,#56,#55,#54,#53,#52,#51,#50, and #49, all begin with "u".

The ones just above those 12 in the list begin with "t".  The last one that
begins with "t" is "tutro", which is #48, and "tutor" is the arrangement just
before that.  Therefore "tutor" is #47 in the list.

Edwin