Question

How many words can be formed using letter of UNIVERSITY using
each letter at most once?
i) If each letter must be used,
ii) If some or all the letters may be omitted.

Answer 1

Letters in the word UNIVERSITY:  E, I,N,R,S,T,U,V,Y

There are 9 distinct letters.   Let us say that each letter is used only once in formation of words of any length.    Let P(m, n) denote the number of permutations of selecting n items from m, ie., m! / (m-n)!

1. Number of 9 letter words:
      If all 9 letters are to be used:    9 !

2.  If words contain  0 to 9 letters:  Sum P(9, n)  with n denoting the number of letters to be present in the word.

         P(9,0) + P(9,1)+ P(9,2) + P(9,3) +.... + P(9,8) + P(9,9)
       (1) + (9) + (9*8)  + (9*8*7) + (9*8*7*6) +.....(9!/3!) + (9! /2! )+ (9!/1!) + 9!
      
please calculated the sum.

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If you are looking for formation of 8 letter words:  P(9,8) = 9!/1! = 9!