An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n numbers s

Q: Why can't we just use $3^n$ for the numerator?

That is the most common trap! $3^n$ counts all sequences formed by your chosen $3$ numbers, but it includes sequences that only use $1$ or $2$ of them. We need *exactly* $3$ distinct numbers, so we must subtract those 'incomplete' sequences.

Q: Why do we add back the $3$ at the end of the formula?

That is the beauty of the Inclusion-Exclusion Principle. When we subtracted the cases where one number was missing, we subtracted the cases where *two* numbers were missing twice. We add them back to correct the count.

Q: What if $n < 3$?

Great observation! If $n < 3$, it is impossible to have $3$ distinct numbers appear. The formula $3^n - 3(2^n) + 3$ will naturally yield $0$ in those cases, which is mathematically consistent.

Question

An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in this list?

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a casino, holding a standard, unbiased die. You are about to throw it $n$ times. Each throw is a moment of pure chance, a tiny event in a sequence of $n$ independent trials.

The total number of possible sequences for $n$ throws is:

$$6^n$$

This represents our denominator, the total universe of outcomes.

### The Exclusive Club

We need exactly $3$ distinct numbers to appear. First, we must decide *which* three numbers will appear out of the $6$ faces on the die.

This is a classic combination problem:

$$\binom{6}{3} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$$

There are $20$ different 'clubs' of three numbers that could potentially appear.

### The Trap of Naivety

Let us fix one such club, say $\{1, 2, 3\}$. We want to fill $n$ slots such that each slot contains one of these three numbers, and each of these three numbers appears at least once.

If we simply say each of the $n$ slots can be filled by any of the $3$ numbers, we get $3^n$ sequences. However, this counts sequences where only one or two numbers appear, which we must filter out.

### The Principle of Inclusion-Exclusion

We use the Principle of Inclusion-Exclusion (PIE) to count sequences where all three numbers appear. Let $A, B,$ and $C$ be the sets of sequences where $1, 2,$ and $3$ are missing, respectively.

We start with $3^n$ and subtract the cases where at least one number is missing:

$$\binom{3}{1} 	imes 2^n = 3 	imes 2^n$$

Because we subtracted the cases where two numbers are missing twice, we must add them back:

$$\binom{3}{2} 	imes 1^n = 3 	imes 1 = 3$$

Thus, the number of valid sequences for a fixed club is $3^n - 3(2^n) + 3$.

### Final Calculation

Since there are $\binom{6}{3}$ such clubs, the total number of favorable outcomes is $\binom{6}{3} 	imes [3^n - 3(2^n) + 3]$. Dividing by the total sample space $6^n$, we arrive at the final probability:

$$P = \frac{\binom{6}{3} [3^n - 3(2^n) + 3]}{6^n}$$

This formula accounts for the selection of the numbers, the arrangement of the sequences, and the rigorous filtering of invalid cases. It is a beautiful example of how complex constraints can be tamed by systematic counting.

Animated Solution for Mathematics - Probability: An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown $n$ times and the list of $n$ numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in this list?

Visualized Solution

Total Possible Outcomes

Sample Space Size

The 'Exactly 3' Constraint

Selecting 3 Numbers

Arranging the Chosen Numbers

The Overcounting Problem

Inclusion-Exclusion Principle

Subtracting 2-Number Sequences

Correcting the Over-subtraction

Valid Sequences per Selection

Final Probability Formula

The Sigma Insight: Classical Definition of Probability

Analyzing the Setup

The Exclusive Club

The Trap of Naivety

The Principle of Inclusion-Exclusion

Final Calculation

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