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Chapter 1: Problem 2
Facing a four-hour bus trip back to college, Diane decides to take along five magazines from the 12 that her sister Ann Marie has recently acquired. In how many ways can Diane make her selection?
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Consider the moves \(\mathrm{R}:(x, y) \rightarrow(x+1, y)\) and \(\mathrm{U}:(x, y) \rightarrow(x, y+1)\), as in Example 1.42. In how many ways can one go a) from \((0,0)\) to \((6,6)\) and not rise above the line \(y=x\) ? b) from \((2,1)\) to \((7,6)\) and not rise above the line \(y=\) \(x-1 ?\) c) from \((3,8)\) to \((10,15)\) and not rise above the line \(y=x+5 ?\)
How many ways are there to place 12 marbles of the same size in five distinct jars if (a) the marbles are all black? (b) each marble is a different color?
Show that for all positive integers \(m\) and \(n\), $$ n\left(\begin{array}{c} m+n \\ m \end{array}\right)=(m+1)\left(\begin{array}{l} m+n \\ m+1 \end{array}\right) $$.
a) Determine the value of the integer variable counter af- ter execution of the following program segment. (Here \(i\), \(j\), and \(k\) are integer variables.) $$ \begin{array}{l}\text { Counter }:=0 \\ \text { for } i:=1 \text { to } 12 \mathrm{do} \\ \text { counter }:=\text { counter }+1 \\ \text { for } j:=5 \text { to } 10 \text { do } \\ \text { counter }:=\text { counter }+2 \\\ \text { for } k:=15 \text { downto } 8 \text { do } \\ \text { counter }:\end{array} $$ counter \(+3\) b) Which counting principle is at play in part (a)?
In the manufacture of a certain type of automobile, four kinds of major defects and seven kinds of minor defects can occur. For those situations in which defects do occur, in how many ways can there be twice as many minor defects as there are major ones?
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