Ten kids line up for recess. The names of the kids are: {Abe, Ben, Cam, Don, Eli, Fran, Gene, Hal, Ike, Jan}. Let S be the set of all possible ways to line up the kids. For example, one ordering might be: (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben) The names are listed in order from left to right, so Fran is at the front of the line and Ben is at the end of the line. Let T be the set of all possible ways to line up the kids in which Gene is ahead of Don in the line. Note that Gene does not have to be immediately ahead of Don. For example, the ordering shown above is an element in T. Now define a function f whose domain is S and whose target is T. Let x be an element of S, so x is one possible way to order the kids. If Gene is ahead of Don in the ordering x, then f(x) = x. If Don is ahead of Gene in x, then f(x) is the ordering that is the same as x, except that Don and Gene have swapped places. (a) What is the output of f on the following input? (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben) (b) What is the output of f on the following input? (Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam) (c) Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer. (d) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, IS| = 3628800. Use this fact and the answer to the previous question to determine |T|. (e) Let Q be the set of orderings in which Gene comes before Don and Jan comes before Abe (again, not necessarily immediately before). Define a k-to-1 correspondence from S to Q. Use the value of k to determine |Q.
Ten kids line up for recess. The names of the kids are: {Abe, Ben, Cam, Don, Eli, Fran, Gene, Hal, Ike, Jan}. Let S be the set of all possible ways to line up the kids. For example, one ordering might be: (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben) The names are listed in order from left to right, so Fran is at the front of the line and Ben is at the end of the line. Let T be the set of all possible ways to line up the kids in which Gene is ahead of Don in the line. Note that Gene does not have to be immediately ahead of Don. For example, the ordering shown above is an element in T. Now define a function f whose domain is S and whose target is T. Let x be an element of S, so x is one possible way to order the kids. If Gene is ahead of Don in the ordering x, then f(x) = x. If Don is ahead of Gene in x, then f(x) is the ordering that is the same as x, except that Don and Gene have swapped places. (a) What is the output of f on the following input? (Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben) (b) What is the output of f on the following input? (Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam) (c) Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer. (d) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, IS| = 3628800. Use this fact and the answer to the previous question to determine |T|. (e) Let Q be the set of orderings in which Gene comes before Don and Jan comes before Abe (again, not necessarily immediately before). Define a k-to-1 correspondence from S to Q. Use the value of k to determine |Q.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Exercise 10.2.5: Using the k-to-1 rule for counting ways to line up a group.
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Ten kids line up for recess. The names of the kids are:
{Abe, Ben, Cam, Don, Eli, Fran, Gene, Hal, Ike, Jan}.
Let S be the set of all possible ways to line up the kids. For example, one ordering might be:
(Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)
The names are listed in order from left to right, so Fran is at the front of the line and Ben is at the end of the line.
Let T be the set of all possible ways to line up the kids in which Gene is ahead of Don in the line. Note that Gene does not have to be
immediately ahead of Don. For example, the ordering shown above is an element in T.
Now define a function f whose domain is S and whose target is T. Let x be an element of S, so x is one possible way to order the kids. If
Gene is ahead of Don in the ordering x, then f(x) = x. If Don is ahead of Gene in x, then f(x) is the ordering that is the same as x, except
that Don and Gene have swapped places.
(a) What is the output of f on the following input?
(Fran, Gene, Hal, Jan, Abe, Don, Cam, Eli, Ike, Ben)
(b) What is the output of f on the following input?
(Eli, Ike, Don, Hal, Jan, Abe, Ben, Fran, Gene, Cam)
(c) Is the function fa k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer.
(d) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, IS| = 3628800. Use this
fact and the answer to the previous question to determine |T|.
(e)
Let Q be the set of orderings in which Gene comes before Don and Jan comes before Abe (again, not necessarily immediately
before). Define a k-to-1 correspondence from Sto Q. Use the value of k to determine |QI.
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