varice2r

2022-09-14

Show that $(n-r){\textstyle (}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{n}{r}{\textstyle )}=n{\textstyle (}\genfrac{}{}{0ex}{}{n+r-1}{2r}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{2r}{r}{\textstyle )}$.

In the LHS $(}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )$ counts the number of ways of selecting $r$ objects from a set of size $n$ where order is not significant and repetitions are allowed. So you have $n$ people you form $r$ teams and select $r$ captains and select $(n-r)$ players.

The RHS divides up a team into $$2$$ sets?

In the LHS $(}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )$ counts the number of ways of selecting $r$ objects from a set of size $n$ where order is not significant and repetitions are allowed. So you have $n$ people you form $r$ teams and select $r$ captains and select $(n-r)$ players.

The RHS divides up a team into $$2$$ sets?

Dalton Erickson

Beginner2022-09-15Added 10 answers

Let $S$ be a set of $n+r-1$ elements. Both sides count the number of ways to select two disjoint sets $A,B\subseteq S$ of size $r$ and possibly an element $c\in S\setminus B$.

We first observe that $(n-r){\textstyle (}\genfrac{}{}{0ex}{}{n}{r}{\textstyle )}=n{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{r}{\textstyle )}$ as both sides count the number of ways to form a team of size $r+1$ with a captain out of $n$ people.

Applying the above on the original LHS we get $n{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{r}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )}$, which corresponds to selecting $A$ ($r$ out of $n+r-1$), then $B$ ($r$ out of the remaining $n-1$) and $c$ ($n-1$ choices in $S\setminus B$ and one option of not choosing $2r$).

The RHS argument goes as follows: choose $2r$ elements for both $A$ and $B$, then choose $r$ of them to make $B$. Then, as before, there are $n$ options of choosing $c\in S\setminus B$ or none at all.

We first observe that $(n-r){\textstyle (}\genfrac{}{}{0ex}{}{n}{r}{\textstyle )}=n{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{r}{\textstyle )}$ as both sides count the number of ways to form a team of size $r+1$ with a captain out of $n$ people.

Applying the above on the original LHS we get $n{\textstyle (}\genfrac{}{}{0ex}{}{n-1}{r}{\textstyle )}{\textstyle (}\genfrac{}{}{0ex}{}{n+r-1}{r}{\textstyle )}$, which corresponds to selecting $A$ ($r$ out of $n+r-1$), then $B$ ($r$ out of the remaining $n-1$) and $c$ ($n-1$ choices in $S\setminus B$ and one option of not choosing $2r$).

The RHS argument goes as follows: choose $2r$ elements for both $A$ and $B$, then choose $r$ of them to make $B$. Then, as before, there are $n$ options of choosing $c\in S\setminus B$ or none at all.

cubanwongux

Beginner2022-09-16Added 4 answers

Dividing the LHS by the RHS and expanding the definition of the choose notation, we have

$$(n-r)\frac{(n+r-1)!}{r!(n-1)!}\frac{n!}{r!(n-r)!}\cdot \frac{1}{n}\frac{(2r)!(n-r-1)!}{(n+r-1)!}\frac{r!r!}{(2r)!}.$$

By cancelling appropriately, it follows that the LHS divided by the RHS is $$1$$, so they're equal.

$$(n-r)\frac{(n+r-1)!}{r!(n-1)!}\frac{n!}{r!(n-r)!}\cdot \frac{1}{n}\frac{(2r)!(n-r-1)!}{(n+r-1)!}\frac{r!r!}{(2r)!}.$$

By cancelling appropriately, it follows that the LHS divided by the RHS is $$1$$, so they're equal.

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix

$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$ Find, correct to the nearest degree, the three angles of the triangle with the given vertices

A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)Whether f is a function from Z to R if

?

a) $f\left(n\right)=\pm n$.

b) $f\left(n\right)=\sqrt{{n}^{2}+1}$.

c) $f\left(n\right)=\frac{1}{{n}^{2}-4}$.How to write the expression ${6}^{\frac{3}{2}}$ in radical form?

How to evaluate $\mathrm{sin}\left(\frac{-5\pi}{4}\right)$?

What is the derivative of ${\mathrm{cot}}^{2}x$ ?

How to verify the identity: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$?

Find $\mathrm{tan}\left(22.{5}^{\circ}\right)$ using the half-angle formula.

How to find the exact values of $\mathrm{cos}22.5\xb0$ using the half-angle formula?

How to express the complex number in trigonometric form: 5-5i?

The solution set of $\mathrm{tan}\theta =3\mathrm{cot}\theta $ is

How to find the angle between the vector and $x-$axis?

Find the probability of getting 5 Mondays in the month of february in a leap year.

How to find the inflection points for the given function $f\left(x\right)={x}^{3}-3{x}^{2}+6x$?

How do I find the value of sec(3pi/4)?