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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.

What is the output of f on the following input?
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)

(b) What is the output of f on the following input?
(Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy)

(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, |S| = 3628800. Use this fact and the answer to the previous question to determine |T|.

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(a)
The input given is
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)

The instructions state that if George is ahead of Dave in the ordering x, then f(x) = x.

Note that this input has George ahead of Dave in the ordering. Since George is ahead of Dave, the output is f(x) = x. That means the output is the same as the input.

Hence, the output is
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby)

(b)
The input given is
(Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy)

The instructions state that if Dave is ahead of George in x, then f(x) is the ordering that is the same as x, ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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  • "Hello, thank you for your answer for my probability question. However, I think you interpreted the second and third question differently than was meant, as the assumption still stands that a person still independently ranks the n options first. The probability I am after is the probability that this independently determined ranking then is equal to one of the p fixed rankings. Similarly for the third question, where the x people choose their ranking independently, and then I want the probability that for x people this is equal to one particular ranking. I was wondering if you could help me with this. "
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