# Here are three ways of approaching the problem 1 i we

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Here are three ways of approaching the problem: 1

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i. We proceed by induction. Note that N 1 = 1 . Observe that we can create a balanced n -by- n matrix by taking a balanced ( n - 1) -by- ( n - 1) matrix and adding an all-zero column and then adding an all-one row. There are n possible placements for the column and n possible placements for the row, so each balanced ( n - 1) -by- ( n - 1) generates n 2 balanced n -by- n matrices. Since every balanced n -by- n can be constructed in this manner, we have N n = N n - 1 · n 2 . It follows that N n = ( n !) 2 . ii. Note that the lower triangular n × n 0-1 matrix is balanced. Moreover, any other balanced matrix may be reduced to this one by rearranging the rows and columns. The total number of such rearrangements is ( n !) 2 . Therefore the answer is ( n !) 2 . iii. There is 1 0-1 row vector of length n with exactly n 1 s. We can place this vector into any of the n possible locations. Next, there are ( n n - 1 ) 0-1 row vectors of length n with exactly n - 1 1 s. We can place any one of these vectors into the any of the remaining n - 1 locations. Note that, after these two choices we specify the column with exactly one 1 , therefore in the next step we can only use ( n - 1 n - 2 ) 0-1 row vectors of length n with exactly n - 2 1 s. The total number of remaining locations is n - 2 . If we proceed similarly, we reduce one degree of freedom after each step, hence we get ( n !) n - 1 Y k =0 n - k n - k - 1 = ( n !) 2 . 4. Given a randomly chosen microprocessor, we define the following events: A : “the microprocessor has been manufactured by A”, B : “the microprocessor has been manufactured by B”, L 10 : “the microprocessor lasts exactly 10 years”, L 15 : “the microprocessor lasts exactly 15 years”, and L 20 : “the microprocessor lasts exactly 20 years”. Note that with this notation, the events “the microprocessor lasts more than 10 years” and “the micro- processor lasts more than 15 years” are represented by L c 10 = ( L 15 L 20 ) and ( L 10 L 15 ) c = L 20 , respectively. We now have P ( A ) = P ( B
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