A word has 4 identical letters and others different. If the total number of words that can be made with the letters of the word be 210 then the number of different letters in the word is

Let the number of different letters be n, meaning the total number of letters is n plus 4. The formula for permutations of n plus 4 letters where 4 are identical is (n plus 4) factorial divided by 4 factorial. Setting this equal to 210 gives (n plus 4) factorial equals 210 multiplied by 24, which is 5040, or 7 factorial. Thus n equals 3.

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