# how many words can be formed from a scrabble rack with 7 letters

Given a scrabble rack with 7 unique letters, how many words (meaning not important) can be formed with 1 to 7 letters?

My first thought was to take all the permutations from p(1,7) to p(7,7) and add them which yielded 13699 but I'm not sure if I'm missing something.

edit: removed the alphabet size as it is of no relevance, and caused some confusion. I'm looking for the total number of words that can be formed with the 7 tiles on the rack.

• Are these only words with length 7, or are shorter words included? Is the empty word valid? – mvw Mar 14 '16 at 20:57
• words of length 1 to 7, empty not valid. – user322826 Mar 14 '16 at 21:01
• Assuming that the language is english, then you have 26 letters so couldn't you just do $26C7 (7P7 + 7P6 + 7P5 + 7P4 + ... + 7P1)$? – Airdish Mar 15 '16 at 15:59
• Your answer of 13699 is correct. – Thanassis Apr 30 at 6:38

Let $\Sigma$ be your alphabet and $\DeclareMathOperator{card}{card}n = \card(\Sigma)$. Then the set of possible words is the language $$L = \bigcup_{k=1}^7 \Sigma^k$$ with $$\card(L) = \sum_{k=1}^7 \card(\Sigma^k) = \sum_{k=1}^7 \binom{n}{k}$$ For $n = 29$ I get $\card(L) = 2\,182\,395$.