Problem: If all possible (distinguishably different) arrangements of the letters A, C, C, L, L, S, U, and U are put in alphabetical order, in what position is the word “CALCULUS”?

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Solution: The total number of distinguishably different arrangements of the letters A, C, C, L, L, S, U, and U is given by

To find the appearance of “CALCULUS,” we first, we need to “clear” all the arrangements that begin with A, of which there are

Those arrangements that begin with “CAC” will appear next on the list and need to be cleared as well. There are

of these, the number of distinguishable arrangements of the remaining letters L, L, S, U, and U.

The next several entries on the list will begin with “CALC,” followed by the letters L, S, U, and U.  We can first clear the arrangements that start with “CALCL,” numbering

and then those that begin “CALCS,” which also number

Then the next arrangement on the list is “CALCULSU,” followed immediately by “CALCULUS.”  Hence, “CALCULUS” appears in position number
630 + 30 + 3 + 3 + 1 + 1 = 668.