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ENGRD2700F11HW2Sol

# ENGRD2700F11HW2Sol - ENGRD 2700 F11 Homework 2 Solutions 1...

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ENGRD 2700 F11 Homework 2 Solutions 1. There are 48 Master of Engineering students to assign to projects this year. Suppose project teams must consist of exactly 4 people. How many diﬀerent ways are there to construct a single project team? What if the team must consist of 2 men and 2 women, and there are 28 men and 20 women students? 1. The number of a single project team is ( 48 4 ) = 194580. 2. For 2 men and 2 women, the number is ( 28 2 )( 20 2 ) = 568. 2. (Based on a real application). A certain oil reﬁnery keeps intermediate products in 8 tanks (tanks are diﬀerent for this question). There are 20 pumps of varying capacity that can be assigned to pump the intermediate product from the 8 tanks into a ﬁnal-product tank. How many ways can you assign the 20 pumps to the 8 tanks (so that each tank gets at least one pump)? To answer this question, we can proceed as follows: Let d ( k,n ) be the number of ways to assign k pumps to n tanks in such a way that each tank gets at least one pump. (a) Explain why d ( k, 1) = 1 for all k > 0. Also explain why d ( k,n ) = 0 for n > k . The reason that d ( k, 1) = 1 for all k > 0 is that the only way to do this is to put all the k pumps in one tank. Also, the reason that d ( k,n ) = 0 for n > k is that for n tanks, there should be at least n pumps, so we cannot ﬁnd a valid partition if n > k . (b) Show that for k > 1 and n 1, d ( k,n ) = nd ( k - 1 ,n ) + nd ( k - 1 ,n - 1) . To partition k pumps into n nonempty tanks, suppose we pick up one pump. We can put this pump into a single tank (there are n diﬀerent tanks) and then partition the remaining k - 1 pumps into n - 1 tanks. Or we can partition the other k - 1 pumps into n tanks and choose one tank to put the pump. From this reasoning, we can get the above equation. (c) Hence or otherwise, answer the question. Feel free to use some software, e.g., Excel.

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ENGRD2700F11HW2Sol - ENGRD 2700 F11 Homework 2 Solutions 1...

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