Simple Combinatorics
Probability
Level
1
If the letters of the word MODESTY are written in all possible orders and these words are all written out as in a dictionary, then the rank of the word MODESTY is
The answer is 1681.
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"First Pass": All words that begin with a D or E will be before MODESTY in the dictionary, e.g. DMOESTY.
Count: (Choice between D or E = 2)*(6! for the remaining letters in any order) = 2 * 6! = 1440
"Second Pass": Any word that begins with an M, but its second letter comes before O, will also be before MODESTY, e.g. MDOESTY.
Count: (Letter M = 1 way to make this choice) * (Choice between D or E = 2) * (5! for the remaining letters in any order) = 2 * 5! = 240
Note that the remaining letters, DESTY, are in alphabetical order already and so there are no ways to have the word start with MO and be before MODESTY alphabetically, so we are done.
Add up all the cases: 1440 + 240 = 1680.
Edit: 1680 is the number of words before MODESTY, so the rank of MODESTY itself is 1680+1 = 1681.