An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers

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

Think about what is changing. The number of choices ($3$) is fixed for every slot, and the number of slots ($n$) is the variable. If you had $3$ slots and $3$ choices, you would have $3 \times 3 \times 3 = 3^3$. The base is the number of options, and the exponent is the number of slots. You have got this!

Q: Does the question imply we can repeat digits? It does not explicitly say 'repetition allowed'.

In combinatorics, unless the problem explicitly states 'without repetition' or 'distinct digits', you must assume repetition is allowed. Since we are forming $n$-digit numbers using a set of digits, we can use '2' in the first slot and '2' again in the second slot. Always look for that 'distinct' keyword—here, it refers to the *numbers* being distinct, not the digits within the number.

Q: Why do we check $n=6$ and $n=7$? Is there a faster way to solve $3^n \ge 900$?

You could use logarithms: $n \ge \log_3(900)$. Since $\log_3(729) = 6$ and $\log_3(2187) = 7$, you would see $n$ must be greater than $6$. But for small integers like this, direct calculation is often faster and less prone to calculation errors than log tables or approximations. Stay grounded in the numbers!

Question

An $n$-digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of $n$ for which this is possible is

Options:

(A) 6

(B) 7

(C) 8

(D) 9

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in front of a digital lock that requires an $n$-digit code. You have only three buttons available: $2$, $5$, and $7$.

Your goal is to create at least $900$ distinct codes. This is not just a math problem; it is a puzzle of possibilities.

### The Slot Machine Analogy

Think of an $n$-digit number as a sequence of $n$ empty slots. Each slot is a decision point.

For the first slot, you have $3$ choices: $2$, $5$, or $7$. For the second slot, you again have $3$ choices.

Because the problem does not forbid repetition, the choice you make for the first slot does not restrict your choices for the second. This independence is the key.

### The Fundamental Principle

The Fundamental Principle of Counting tells us that if one event can occur in $m$ ways and a second independent event can occur in $k$ ways, then the two events together can occur in $m 	imes k$ ways.

Extending this to $n$ slots, we multiply the number of choices for each slot: $3 	imes 3 	imes \dots 	imes 3$ ($n$ times). This gives us a total of $3^n$ distinct numbers.

### The Search for $n$

Now, we face the inequality:

$$3^n \ge 900$$

We need the smallest integer $n$ that satisfies this. Let us test our values systematically.

For $n=6$, we calculate:

$$3^6 = 729$$

Since $729 < 900$, this is not enough. We need more slots.

For $n=7$, we calculate:

$$3^7 = 3^6 	imes 3 = 729 	imes 3 = 2187$$

Since $2187 \ge 900$, we have finally crossed the threshold.

The smallest integer $n$ is **$7$**. It is a beautiful result—a reminder that even with a small set of digits, the power of exponents allows us to generate a vast array of possibilities.

Animated Solution for Mathematics - Permutations and Combinations: An $n$ -digit number is a positive number with exactly $n$ digits. Nine hundred distinct $n$ -digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of $n$ for which this is possible is

Select Answer:

Visualized Solution

Understanding the $n$ -digit structure

Choices for the first slot

Choices for subsequent slots

Choices for all $n$ slots

Total number of $n$ -digit numbers

Setting the condition

Evaluating powers: $n = 6$

Evaluating powers: $n = 7$

Conclusion

The Sigma Insight: Fundamental Principle of Counting

Analyzing the Setup

The Slot Machine Analogy

The Fundamental Principle

The Search for $n$

Similar Questions

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

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is

The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9 (repetition of digits allowed) is equal to :

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ......... .

The number of five-digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is :

If number of 5 digit numbers which can be formed without repeating any digit while tenth place of all of the numbers must be 2 is 336 k find value of k

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5 is

If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 1 0^{n}$ ; $m, n \in {1, 2, 3, \dots, 9}$ , then $m + n$ is equal to _______