Post History
#2: Post edited
by
Wolgwang
·
2021-09-02T14:11:53Z (over 2 years ago)
Mathjax error fixed
Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"?
- Because you "first choose the k team members and then choose one of time to be captain", shouldn’t the RHS be $\color{red}{k\dbinom{k}{1}}$? The captain is chosen from the $k$ team members already chosen.
- $\color{forestgreen}{k\dbinom{n}{k}}$ appears wrong to me, because this means that you're choosing the captain from the original group of $n$ people.
- >### Example 1.5.2 (The team captain).
- For any positive integers n and k with $k \le n$, $n\dbinom{n - 1}{k - 1} = \color{forestgreen}{k\dbinom{n}{k}}$.
>This is again easy to check algebraically (using the fact that $m! = m(m - 1)! for- any positive integer $m$), but a story proof is more insightful.
- >Story proof : Consider a group of n people, from which a team of k will be chosen,
- one of whom will be the team captain. To specify a possibility, we could first choose
- the team captain and then choose the remaining $k - 1$ team members; this gives
- the left-hand side. Equivalently, we could fi rst choose the k team members and then
- choose one of them to be captain; this gives the right-hand side. $\square$
- Blitzstein. *Introduction to Probability* (2019 2 ed). Pages 20-21.
- Because you "first choose the k team members and then choose one of time to be captain", shouldn’t the RHS be $\color{red}{k\dbinom{k}{1}}$? The captain is chosen from the $k$ team members already chosen.
- $\color{forestgreen}{k\dbinom{n}{k}}$ appears wrong to me, because this means that you're choosing the captain from the original group of $n$ people.
- >### Example 1.5.2 (The team captain).
- For any positive integers n and k with $k \le n$, $n\dbinom{n - 1}{k - 1} = \color{forestgreen}{k\dbinom{n}{k}}$.
- >This is again easy to check algebraically (using the fact that $m! = m(m - 1)!$ for
- any positive integer $m$), but a story proof is more insightful.
- >Story proof : Consider a group of n people, from which a team of k will be chosen,
- one of whom will be the team captain. To specify a possibility, we could first choose
- the team captain and then choose the remaining $k - 1$ team members; this gives
- the left-hand side. Equivalently, we could fi rst choose the k team members and then
- choose one of them to be captain; this gives the right-hand side. $\square$
- Blitzstein. *Introduction to Probability* (2019 2 ed). Pages 20-21.
#1: Initial revision
by
DNB
·
2021-07-09T06:38:52Z (almost 3 years ago)
Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"?
Because you "first choose the k team members and then choose one of time to be captain", shouldn’t the RHS be $\color{red}{k\dbinom{k}{1}}$? The captain is chosen from the $k$ team members already chosen.
$\color{forestgreen}{k\dbinom{n}{k}}$ appears wrong to me, because this means that you're choosing the captain from the original group of $n$ people.
>### Example 1.5.2 (The team captain).
For any positive integers n and k with $k \le n$, $n\dbinom{n - 1}{k - 1} = \color{forestgreen}{k\dbinom{n}{k}}$.
>This is again easy to check algebraically (using the fact that $m! = m(m - 1)! for
any positive integer $m$), but a story proof is more insightful.
>Story proof : Consider a group of n people, from which a team of k will be chosen,
one of whom will be the team captain. To specify a possibility, we could first choose
the team captain and then choose the remaining $k - 1$ team members; this gives
the left-hand side. Equivalently, we could fi rst choose the k team members and then
choose one of them to be captain; this gives the right-hand side. $\square$
Blitzstein. *Introduction to Probability* (2019 2 ed). Pages 20-21.