Minimum number of alphabets required in a language to create at least 1 million distinct three letter initials

Introduction / Context:
This problem relates to counting the number of possible three letter initials you can form from an alphabet of a given size. It is similar to counting license plates or codes and tests your understanding of basic permutation counting with repetition allowed.

Given Data / Assumptions:

• We want to form three letter initials of the form X Y Z.
• Each position in the initial can be any letter from the alphabet.
• Letters can repeat in an initial.
• We need at least 1 million distinct three letter initials.
• The alphabet size, say n, is the same for each position.

Concept / Approach:
When each of the 3 positions can independently be any of n letters, the total number of distinct three letter strings is n^3. The requirement is that n^3 should be at least 1 million. We therefore find the smallest integer n such that n^3 is greater than or equal to 1000000.

Step-by-Step Solution:
Step 1: Let n be the number of different letters in the alphabet.Step 2: The number of possible three letter initials is n^3 because each of the 3 positions can be filled in n ways.Step 3: We need n^3 greater than or equal to 1000000.Step 4: Observe that 100^3 = 100 * 100 * 100 = 1000000 exactly.Step 5: For any integer n less than 100, such as 99, the cube n^3 is less than 1000000.Step 6: Therefore the smallest integer n that satisfies n^3 greater than or equal to 1000000 is 100.

Verification / Alternative check:
You can compute 10^3 = 1000 and 50^3 = 125000, which are far below 1 million. Checking 100^3 gives exactly 1 million, confirming that this alphabet size is just enough to reach the requirement.

Why Other Options Are Wrong:

• 1000 would give 1000^3, which is 1000 billion, far more than needed; it is not the minimum.
• 500 would give 125 million three letter initials, again too large and not minimal.
• 999 is also unnecessarily large and would exceed the requirement by a huge margin.

Common Pitfalls:
Some learners misinterpret the requirement and think that the number of initials must be exactly equal to 1 million rather than at least 1 million. Others mistakenly use nC3 instead of n^3, which would be appropriate only if letters could not repeat and order did not matter. Here, order clearly matters and repetition is allowed.

Final Answer:
The minimum alphabet size required is 100 letters, so the correct answer is 100.