How many symbols must an alphabet contain to form exactly 1 million possible…
2023
How many symbols must an alphabet contain to form exactly 1 million possible distinct three-letter initials, assuming repetition is allowed?
Answer: D. 100 — ConceptFor a sequence of n positions, if each position independently allows A symbols and repetition is allowed, the multiplication principle gives An…
- A.
140
- B.
1000
- C.
10
- D.
100
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Correct answer: D
Concept
For a sequence of n positions, if each position independently allows A symbols and repetition is allowed, the multiplication principle gives An possible ordered sequences.
To recover A from a specified total N, take the nth root of N.
Application
Let A be the number of symbols in the alphabet.
There are three independent positions, so the number of initials is A × A × A = A3.
Set A3 = 1,000,000 = 106.
Taking the cube root gives A = ∛(106) = 102 = 100.
Cross-check
Substitution gives 1003 = 1,000,000, exactly the required number of distinct ordered initials.
Therefore, the alphabet must contain 100 symbols.