This post might interest Scrabble aficionados. They would certainly have played games in which all the one hundred tiles have been used up. And instances may be numerous where a bingo (making a seven-letter word) or two have been scored.
This is something I found in my scrapbook. I am not sure if I 'invented' it during one of those lonely Sundays afternoons in Calcutta in the early 1970’s when I had nothing else to do. It is equally possible that I copied it from somewhere. (The latter is more likely, given that I am not all that inventive.)
Have a look at this formation:
Unfortunately, I do not have a clearer picture than this.
The interesting feature about this grid is that all the letters in a standard Scrabble game have been used. The blanks have been used as I and A. Formation of the words has been done strictly complying with all the standard rules prescribed for forming words. This is theoretically possible if the appropriate letters happen to be picked up by the players in succession.
A possible order in which the words are formed is:
QUEUING, BEHEADED, AQUARIUM, WEAKLING, DELAYING, ABILITY & YA, REVIVING, OOZIEST & GO, ISOLATOR, COMPACT & IT, SWEATER & AQUARIUMS, PONTIFFS, EXHORTER, JOINDER & ER, IS and AS (or US - using the blank).
As you can see, there are fourteen words of seven or more letters. I guess the highest possible number of seven-letter words have been formed.
The only problem is that the probability of two players picking up the tiles required for this formation is zero. I know it is not zero, but those who say it is not will have to calculate and tell me the exact probability.
As there are 100 tiles consisting, inter alia, of three B's, twelve E's, three H's, and four D's, the probability of the first player picking up one E, one G, one N, one Q, two U's and one blank (not necessarily I that order) from the one hundred tiles would, if I am not mistaken, be (12/100)*(3/99)*(6/98)*(1/97)*(4/96)*(3/95)*(2/94) = 0.000 000 000 064 255 350 419 873 374 551 141 572 335 707. (Till someone proves me wrong,I will stick to this number!)