Ten kids line up for recess. The names of the kids are: {Alex, Bobby, Cathy, Dave, Emy, Frank, George, Homa, Ian, Jim}. Let S be the set of all possible ways to line up the kids. For example, one order might be: (Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby) The names are listed in order from left to right, so Frank is at the front of the line and Bobby is at the end of the line. Let T be the set of all possible ways to line up the kids in which George is ahead of Dave in the line. Note that George does not have to be immediately ahead of Dave. 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 George is ahead of Dave in the ordering x, then f(x) = x. If Dave is ahead of George in x, then f(x) is the ordering that is the same as x, except that Dave and George have swapped places. (1) Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer. (2) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, |S| = 3628800. Use this fact and the answer to the previous question to determine |T|.
Ten kids line up for recess. The names of the kids are:
{Alex, Bobby, Cathy, Dave, Emy, Frank, George, Homa, Ian, Jim}.
Let S be the set of all possible ways to line up the kids. For example, one order might be:
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)
The names are listed in order from left to right, so Frank is at the front of the line and Bobby is at the end of the line.
Let T be the set of all possible ways to line up the kids in which George is ahead of Dave in the line. Note that George does not have to be immediately ahead of Dave. 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 George is ahead of Dave in the ordering x, then f(x) = x. If Dave is ahead of George in x, then f(x) is the ordering that is the same as x, except that Dave and George have swapped places.
(1) Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer.
(2) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, |S| = 3628800. Use this fact and the answer to the previous question to determine |T|.
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