If a monster has 63 children and he wants to keep 3 letter names for each of them so that they are distinct,but with the condition that you can use the same letter more than once,how many letters at minimum does the monster need to name it's children?-source-BdMO 2011question.I have done a bit of listing and I think it is 5. As others pointed out,my listing was wrong.
Let $n$ be the number of letters we need to form $63$ distinct three-letter names.
So, we have $n$ choices for the first letter, $n$ choices for the second letter and $n$ choices for the third letter: This means we have $n\times n\times n = n^3$ possible combinations of distinct names.
We need $n^3 \geq 63$ where $n$ is an integer. $n = 4$ gives us $4^3 = 64$ distinct names, and this is indeed the least integer giving us at least $63$ distinct names. (For $n = 3$, we have only $3^3 = 27$ possible distinct names, and for $n = 5$, we have as many as $5^3 = 125$ distinct three letter names.
Hence, the monster needs minimally $4$ distinct letters to create 63 distinct three-letter names.
On the assumption that order matters, so you can use both aab and aba, for $n$ letters there are $n^3$ names and we need $n=4$.
If order does not matter, we have $n$ names with all letters the same, $n(n-1)$ with two of one letter and one of another, and $\frac 16n(n-1)n-2$ with all letters different. This adds up to $\frac 16n^3+\frac 12n^2+\frac 13n$ and we need $n=7$
The answers listed above have attached an unspecified paradigm - i.e.visual form of the names. The Question states that this is a monster, not a man, and the criteria is "distinction". Why do we seek an answer in the written form? Is this a known characteristic of monsters?
Webster - DISTINCTION : a difference that you can see, hear, smell, feel, etc. : a noticeable difference between things or people
Wouldn't 3 letters, pronounced at different octaves, or in ascending tones, or descending tones, etc. meet the test of distinction? and since letters may be repeated..........
My answer is 1 letter