start n11 n 12 n21 n22 n23 n31 n32 133 n34 n41 142 n43 144 145 n51 n52 153 154 761 162 n63 n71 n72 end Figure 2: A bigger DAG. 1.3 Exercise 1. Show that the number of paths from start to end in the DAG in Figure 2 is 70. 2. Let us number the rows in the DAG starting from 0. So the number of the last row is 8. We could of course make larger figures in an exactly similar way, containing more rows. In any case the number of the last row would be an even number. (This should be pretty clear, I hope.) Let us denote the number of the last rows by 2k. Find an expression for the number of paths from start to end in such a DAG where 2k is the number of the last row. 3. Show that the number of such paths is CR (You will need Stirling's formula to show this. In fact, you'll need one of the strengthened forms at the end of the Stirling's formula handout.) start n11 n 12 n21 n22 n23 n31 n32 133 n34 n41 142 n43 144 145 n51 n52 153 154 761 162 n63 n71 n72 end Figure 2: A bigger DAG. 1.3 Exercise 1. Show that the number of paths from start to end in the DAG in Figure 2 is 70. 2. Let us number the rows in the DAG starting from 0. So the number of the last row is 8. We could of course make larger figures in an exactly similar way, containing more rows. In any case the number of the last row would be an even number. (This should be pretty clear, I hope.) Let us denote the number of the last rows by 2k. Find an expression for the number of paths from start to end in such a DAG where 2k is the number of the last row. 3. Show that the number of such paths is CR (You will need Stirling's formula to show this. In fact, you'll need one of the strengthened forms at the end of the Stirling's formula handout.)

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start n11 n 12 n21 n22 n23 n31 n32 133 n34 n41 142 n43 144 145 n51 n52 153 154 761 162 n63 n71 n72 end Figure 2: A bigger DAG. 1.3 Exercise 1. Show that the number of paths from start to end in the DAG in Figure 2 is 70. 2. Let us number the rows in the DAG starting from 0. So the number of the last row is 8. We could of course make larger figures in an exactly similar way, containing more rows. In any case the number of the last row would be an even number. (This should be pretty clear, I hope.) Let us denote the number of the last rows by 2k. Find an expression for the number of paths from start to end in such a DAG where 2k is the number of the last row. 3. Show that the number of such paths is CR (You will need Stirling's formula to show this. In fact, you'll need one of the strengthened forms at the end of the Stirling's formula handout.) start n11 n 12 n21 n22 n23 n31 n32 133 n34 n41 142 n43 144 145 n51 n52 153 154 761 162 n63 n71 n72 end Figure 2: A bigger DAG. 1.3 Exercise 1. Show that the number of paths from start to end in the DAG in Figure 2 is 70. 2. Let us number the rows in the DAG starting from 0. So the number of the last row is 8. We could of course make larger figures in an exactly similar way, containing more rows. In any case the number of the last row would be an even number. (This should be pretty clear, I hope.) Let us denote the number of the last rows by 2k. Find an expression for the number of paths from start to end in such a DAG where 2k is the number of the last row. 3. Show that the number of such paths is CR (You will need Stirling's formula to show this. In fact, you'll need one of the strengthened forms at the end of the Stirling's formula handout.)

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start n11 n 12 n21 n22 n23 n31 n32 133 n34 n41 142 n43 144 145 n51 n52 153 154 761 162 n63 n71 n72

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