A matrix with four rows and three columns is to be formed with entries -, 1 or 2. How many such distinctmatrieces are possible?
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Hint: How many 1's must each row contain? And thus how many 0's? And what about each column?
Based on this, can you come up with a combinatorial formula for counting the possible ways in which the 0's can be placed in the matrix? (Hint #2: Think about permutations.)
OK, let me be a little more explicit. I assume you've already figured out that each row must contain exactly one zero. We can place that zero in any column we want — but once we do, we cannot put any more zeros in that column.
So, for the first row, there are five possible columns we can put the zero in. For the second row, only four possible columns remain. For the third row, there are only three columns we can place the zero in, and so on.
Based on this, can you come up with a combinatorial formula for counting the possible ways in which the 0's can be placed in the matrix? (Hint #2: Think about permutations.)
OK, let me be a little more explicit. I assume you've already figured out that each row must contain exactly one zero. We can place that zero in any column we want — but once we do, we cannot put any more zeros in that column.
So, for the first row, there are five possible columns we can put the zero in. For the second row, only four possible columns remain. For the third row, there are only three columns we can place the zero in, and so on.
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