QUANTITY OF VARIETY 7/6 



which contains twelve elements, contains only three distinct elements 

 — a, b and c. Such a set will be said to have a variety of three ele- 

 ments. (A qualification is added in the next section.) 



Though this counting may seem simple, care is needed. Thus 

 the two-armed semaphore can place each arm, independently of 

 the other, in any of eight positions; so the two arms provide 64 

 combinations. At a distance, however, the arms have no indi- 

 viduality — "arm A up and arm B down" cannot be distinguished 

 from "arm A down and arm B up" — so to the distant observer only 

 36 positions can be distinguished, and the variety is 36, not 64. It 

 will be noticed that a set's variety is not an intrinsic property of the 

 set: the observer and his powers of discrimination may have to be 

 specified if the variety is well defined. 



Ex. 1 : With 26 letters to choose from, how many 3-letter combinations are 

 available for motor registration numbers ? 



Ex. 2: If a farmer can distinguish 8 breeds of chicks, but camiot sex them, while 

 his wife can sex them but knows nothing of breeds, how many distinct classes 

 of chicks can they distinguish when working together ? 



Ex. 3 : A spy in a house with four windows arranged rectangularly is to signal 

 out to sea at night by each window showing, or not showing, a light. How 

 many forms can be shown if, in the darkness, the position of the lights 

 relative to the house cannot be perceived? 



Ex. 4: Bacteria of different species differ in their ability to metabolise various 

 substances: thus lactose is destroyed by E. coli but not by E. typhi. If a 

 bacteriologist has available ten substances, each of which may be destroyed 

 or not by a given species, what is the maximal number of species that he 

 can distinguish? 



Ex. 5 : If each Personality Test can distinguish five grades of its own character- 

 istic, what is the least number of such tests necessary to distinguish the 

 2,000,000,000 individuals of the world's population? 



Ex. 6: In a well-known card trick, the conjurer identifies a card thus: He shows 

 21 cards to a by-stander, who selects, mentally, one of them without re- 

 vealing his choice. The conjurer then deals the 21 cards face upwards 

 into three equal heaps, with the by-stander seeing the faces, and asks him 

 to say which heap contains the selected card. He then takes up the cards, 

 again deals them into three equal heaps, and again asks which heap contains 

 the selected card; and similarly for a third deal. The conjurer then names 

 the selected card. What variety is there in (i) the by-stander's indications, 

 (ii) the conjurer's final selection ? 



Ex. 7: (Continued.) 21 cards is not, in fact, the maximal number that could 

 be used. What is the maximum, if the other conditions are unaltered ? 



Ex. 8: (Continued.) How many times would the by-stander have to indicate 

 which of three heaps held the selected card if the conjurer were finally to 

 be able to identify the correct card out of the full pack of 52 ? 



Ex. 9: If a child's blood group is O and its mother's group is O, how much 

 variety is there in the groups of its possible fathers ? 



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