Animated Solution for Mathematics - Permutations and Combinations: The number of matrices of order $3\times 3$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is

Unlocking the $3\times 3$ Matrix

Imagine you are standing before a blank $3\times 3$ grid. You have nine empty cells, and in your hand, you hold a bag of binary digits—zeros and ones.

Your task is to fill this grid such that the total sum of the digits is a prime number. In the world of JEE Advanced, the difference between a good student and a great one is the ability to break a complex constraint into manageable, elegant cases.

Defining the Universe of Constraints

A $3\times 3$ matrix has exactly $3\times 3=9$ entries. If we denote each entry as $a_{ij}$, where $a_{ij}\in \{0,1\}$, the sum of all entries, $S$, is simply the count of how many '1's we have placed in the grid.

Since we have 9 slots, the minimum sum is 0 and the maximum sum is 9. Thus, $S\in \{0,1,2,\dots ,9\}$.

Our condition is that $S$ must be a prime number. Looking at our set, the primes are $\{2,3,5,7\}$.

Note that 1 is not prime, and 0 is not prime. Never let a simple definition trip you up in the heat of an exam!

The Art of Combinatorial Selection

To find the number of matrices for a given sum $S$, we are essentially asking: "In how many ways can I choose $S$ positions out of 9 to place a 1?" This is the classic definition of a combination: $\left(\genfrac{}{}{0ex}{}{9}{S}\right)$.

We must calculate this for each of our prime candidates using the formula:

The Sum is 2 We need to choose 2 positions out of 9:

The Sum is 3 Now we choose 3 positions out of 9:

The Sum is 5 Here, we use the symmetry property $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-r}\right)$. Calculating $\left(\genfrac{}{}{0ex}{}{9}{5}\right)$ is the same as $\left(\genfrac{}{}{0ex}{}{9}{4}\right)$:

The Sum is 7 Finally, we choose 7 positions out of 9, which is equivalent to choosing 2 positions to be '0':

Since these cases are mutually exclusive, we simply add the results of our individual cases to find the total number of valid matrices:

The value 282 represents every possible configuration of 0s and 1s that satisfies our prime sum condition.

When you approach problems like this, visualize the grid, define your constraints clearly, and move through the cases with the precision of a surgeon. You have just mastered a fundamental principle of combinatorics.