n-digit numbers are formed using only three digits 2,5 and 7. The smallest value of n for which 900

Q: Wait, why is it $3^n$ and not $n^3$? I always get the base and the exponent mixed up.

That is a classic point of confusion! Remember: the base is the number of choices you have for each individual slot, and the exponent is the number of slots you are filling. Since you have $3$ choices for every one of the $n$ slots, you multiply $3$ by itself $n$ times, which gives $3^n$.

Q: Does the order of digits matter here? What if I swap a 2 and a 5?

In the context of forming numbers, the position is everything! The number $25$ is fundamentally different from $52$. Because each slot is distinct, swapping digits creates a new unique number, which is exactly why we use the multiplication principle to count all possible permutations.

Q: Why can't we just use logarithms to solve $3^n \ge 900$?

You absolutely can! Using logarithms, $n \ge \log_3(900)$. Since $\log_3(729) = 6$ and $\log_3(2187) = 7$, you would find $n \ge 6.something$. Since $n$ must be an integer, you round up to $7$. Both methods are mathematically sound, but for small powers, direct calculation is often faster and less prone to error.

Q: Is there any restriction on using the same digit multiple times?

The problem states we are forming $n$-digit numbers using the digits $2, 5,$ and $7$. It does not say 'without repetition,' so we assume we have an infinite supply of each digit for every slot. This is why each slot remains independent with $3$ choices.

Q: Why did we check $n=6$ and $n=7$ specifically?

We checked those because we needed to find the *smallest* integer $n$ that crosses the threshold of $900$. By testing values around the expected range, we confirm that $6$ falls short while $7$ comfortably exceeds the requirement, ensuring $7$ is indeed the minimum.

Question

n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is

Options:

(A) 9

(B) 6

(C) 8

(D) 7

Sigma Percentile · Accepted Answer

### The Architecture of Combinations: Building Numbers from Scratch

Imagine you are an architect, but instead of steel and glass, you are building numbers. You have a limited palette of materials: the digits $2$, $5$, and $7$.

Your task is to construct an $n$-digit number. This isn't just a math problem; it is a study in the power of exponential growth. When we talk about $n$-digit numbers, we are essentially talking about $n$ empty slots waiting to be filled.

### The Fundamental Principle of Counting

Let us look at the first slot. You have three distinct choices: $2$, $5$, or $7$. That is $3$ ways to start your number.

Now, move to the second slot. Does the choice you made for the first slot restrict your second choice? No. The problem implies that each position is independent. Therefore, for the second slot, you again have $3$ choices.

If you have $3$ choices for the first slot and $3$ for the second, the total number of combinations for a $2$-digit number is $3 	imes 3 = 3^2 = 9$. As we extend this to $n$ slots, the logic remains beautifully consistent.

By the Fundamental Principle of Counting, the total number of distinct $n$-digit numbers is the product of the choices for each slot:

$$Total = 3 	imes 3 	imes 3 	imes \dots 	imes 3 	ext{ ($n$ times)} = 3^n$$

This expression, $3^n$, is the heartbeat of our problem. It represents the explosive growth of possibilities as we add just one more digit to our sequence.

### The Threshold of 900

We are tasked with finding the smallest $n$ such that we can form at least $900$ distinct numbers. Mathematically, we are solving the inequality:

$$3^n \ge 900$$

This is where the "thrill of the hunt" begins. We need to find the tipping point. Let us test our powers of $3$:

$$3^1 = 3$$
$$3^2 = 9$$
$$3^3 = 27$$
$$3^4 = 81$$
$$3^5 = 243$$

We are getting closer, but we are still well below our target of $900$.

### The Moment of Truth

Now, let us look at $n = 6$. Calculating $3^6$, we get:

$$3^6 = 3^5 	imes 3 = 243 	imes 3 = 729$$

Pause for a moment. $729$ is close to $900$, but it is not enough. If we only had $6$ slots, we could only form $729$ unique numbers. We need at least $900$, which means $n=6$ is insufficient.

When we step up to $n = 7$, we calculate:

$$3^7 = 729 	imes 3 = 2187$$

Suddenly, we have vaulted past our requirement of $900$. Since $2187 \ge 900$, we have found our answer. The smallest integer $n$ that satisfies our condition is **$7$**.

### Reflecting on the Elegance

It is fascinating to see how quickly the numbers grow. At $n=6$, we were short by nearly $200$ combinations. By simply adding one more slot—one more digit to our sequence—we more than doubled our capacity, jumping from $729$ to $2187$.

This is the beauty of exponential functions. They start slow, but they possess a hidden, overwhelming power. You have successfully navigated the logic of permutations and the growth of powers. Keep this intuition for your next challenge; the ability to visualize these 'slots' is a tool that will serve you well in every combinatorics problem you face.

Animated Solution for Mathematics - Permutations and Combinations: n-digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is

Select Answer:

Visualized Solution

Visualizing $n$ Slots

Choices for the First Slot

Choices for the Second Slot

Generalizing to $n$ Slots

The Multiplication Principle

Setting the Condition

Testing $n = 6$

Testing $n = 7$

Final Conclusion

The Sigma Insight: Fundamental Principle of Counting

The Architecture of Combinations

The Fundamental Principle of Counting

The Threshold of 900

The Moment of Truth

Reflecting on the Elegance

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