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Sudoku Proof Jumbles, We are going to show there are \(4^{16}\) ways to…

- - - - Now, let's take a look at \(a_{11}\).
        
        Since the possible numbers in our set are \(\{1,2,3,4\}\), \(a_{11}\) has four options.
        
        The same applies for \(a_{12}\); there are four options.
        
        In general, each \(a_{ij}\) has four options, where \(i\) represents the row position and \(j\) represents the column position.
        
        Since there are 16 positions in our Matrix A, \(4 \cdot 4 \cdot 4 \cdot \dots \cdot 4 \) (16 times) = \(4^{16}\).
        
        Therefore there are \(4^{16}\) ways to fill a 4 by 4 grid with no restrictions.
        
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- - - - The 2 by 2 regions we are focusing on are Region TL (top left) = \(\{ a_{11}, a_{12}, a_{21}, a_{22} \} \), Region TR (top right) = \(\{ a_{13}, a_{14}, a_{23}, a_{24} \} \), Region BL (bottom left) = \(\{ a_{31}, a_{32}, a_{41}, a_{42} \} \), and Region BR (bottom right) = \(\{ a_{33}, a_{34}, a_{43}, a_{44} \} \)
        
        Now, let's take a look at Region TL (top left) = \(\{ a_{11}, a_{12}, a_{21}, a_{22} \} \). We know elements 1 through 4 are used exactly once in Region TL. Let's start with \(a_{11}\). If we let \(a_{11}\) have four options, then, that element cannot be in \(a_{12}, a_{21}\), or \(a_{22}\). Therefore, when \(a_{11}\) uses one of the elements, there are three remaining elements.
        
        Next, let's take a look at \(a_{12}\). Since an element was placed in \(a_{11}\), we know there are three remaining elements. When an element is placed in \(a_{12}\) there will be two elements left to be placed in \(a_{21}\) and \(a_{22}\).
        
        Now, let's take a look at \(a_{21}\). Since \(a_{11}\) and \(a_{12}\) already have two of our fours elements, we know \(a_{21}\) has two elements as options. After one of the elements is placed in \(a_{21}\), there will be one element remaining. We know this remaining element will be placed in \(a_{22}\). Therefore, \(a_{22}\) has one element option.
        
        For Region TL we have shown \(a_{11}\) has 4 element options, \(a_{12}\) has 3 element options, \(a_{21}\) has 2 element options, and \(a_{22}\) has 1 element option.
        
        In other words, a 2 by 2 region has 4! ways to fill the region using the elements 1 through 4 only once.
        
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