Animated Solution for Mathematics - Permutations and Combinations: The number of 5-digit natural numbers, such that the product of their digits is 36, is ______.

My dear student, welcome to the fascinating world of combinatorics. Today, we are not just solving a math problem; we are acting as detectives.

We are given a constraint: a 5-digit number whose digits multiply to 36. This is a classic JEE Advanced style problem because it tests your ability to organize chaos. It is not about brute force; it is about systematic decomposition.

The Prime Factorization

Before we even think about the 5-digit number, we must understand the building blocks. The prime factorization of 36 is:

This tells us that any 5-digit number satisfying our condition must be constructed using these four prime factors: two $2$s and two $3$s.

But wait, we have 5 slots to fill! This is where the digit $1$ becomes our best friend. Since multiplying by $1$ does not change the product, we can use as many $1$s as we need to fill the remaining slots.

The Systematic Hunt

We must be methodical. We cannot just guess; we must build our cases based on the largest possible digits we can form.

Case 1: Using a $9$ ($3\times 3$). The remaining factors are $2\times 2=4$. Our digits are $\{9,4,1,1,1\}$. The number of arrangements is:

Case 2: Using a $9$ and splitting the $4$ into $2\times 2$. Our digits are $\{9,2,2,1,1\}$. The arrangements are:

Case 3: Forming two $6$s ($2\times 3$ and $2\times 3$). Our digits are $\{6,6,1,1,1\}$. The arrangements are:

Case 4: Forming one $6$ ($2\times 3$), leaving $3$ and $2$ separate. Our digits are $\{6,3,2,1,1\}$. The arrangements are:

Case 5: Forming no $6$s, but combining the $2$s into a $4$. Our digits are $\{4,3,3,1,1\}$. The arrangements are:

Case 6: Keeping all prime factors separate. Our digits are $\{3,3,2,2,1\}$. The arrangements are:

The Grand Total

We have systematically exhausted every possible way to partition the prime factors of 36 into 5 slots. We have accounted for every repetition, every combination, and every possibility.

The total number of such 5-digit numbers is the sum of all our permutations:

There you have it! The final answer is 180 distinct numbers. This problem teaches us that in combinatorics, the answer is found in the elegance of your case-building.