Combinatorial Proof I have trouble coming up with combinatorial proofs. How would you justify this equality?
$$
n\binom {n-1}{k-1} = k \binom nk
$$
where $n$ is a positive integer and $k$ is an integer.
 A: Say we want to choose a baseball team of $k$ players from $n$ players and we want to choose a captain for the baseball team. We can count the above in two different ways.
$1$. Choose the captain first. This can be done in $n$ ways. Now choose the rest of the team, i.e., we need to choose $k-1$ people from the remaining $n-1$ people, which can be done in $\dbinom{n-1}{k-1}$ ways.
$2$. Choose the team first, i.e., choose $k$ players from $n$ players. This can be done in $\dbinom{n}k$ ways. Now once we have these $k$ players, the captain can be chosen in $k$ ways.
A: We have a group of $n$ people, and want to count the number of ways to choose a committee of $k$ people with Chair. 
For the left-hand side, we select the Chair first, and then $k-1$ from the remaining $n-1$ to join her.
For the right-hand side, we choose $k$ people, and select one of them to be Chair. 
A: I would write the expression for the binomial coefficients in terms of factorials and notice what happens.
A: Since k is on the right-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . , switch the sides so it is on the left-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . . 
Multiply
multiply
To compute a product; to perform a multiplication.  n by each term
term
Any expression written as a product or quotient. Example: 2xy ,  , or  inside the parentheses
parentheses
Marks of inclusion (symbols: ( and ) ). Parentheses is the plural form of parenthesis. .  
Multiply
multiply
To compute a product; to perform a multiplication.  k by each term
term
Any expression written as a product or quotient. Example: 2xy ,  , or  inside the parentheses
parentheses
Marks of inclusion (symbols: ( and ) ). Parentheses is the plural form of parenthesis. .  
Since k is on the right-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . , switch the sides so it is on the left-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . . 
Since the variable
variable
A letter used to represent a number value in an expression or an equation.  is in the denominator
denominator
The bottom part of a fraction.  on the left-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . , this can be solved as a ratio
ratio
A pair of numbers that compares different types of units. . For example,  is equivalent
equivalent
Two or more expressions that have the same value.  to 
Since k is on the right-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . , switch the sides so it is on the left-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = . .  
Reduce the expression
expression
A mathematical symbol, or combination of symbols, representing a value, or relation. Example: 2+2=4  by canceling out all common factors
factor
One of two or more expressions that are multiplied together to get a product.  from the numerator
numerator
The top part of a fraction.  and denominator
denominator
The bottom part of a fraction. . 
k-1=(n-1) 
Remove the parentheses
parentheses
Marks of inclusion (symbols: ( and ) ). Parentheses is the plural form of parenthesis.  around the expression
expression
A mathematical symbol, or combination of symbols, representing a value, or relation. Example: 2+2=4  n-1. 
k-1=n-1 
Since -1 does not contain the variable
variable
A letter used to represent a number value in an expression or an equation.  to solve for, move it to the right-hand side of the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = .  by adding 1 to both sides. 
k=1+n-1 
Subtract 1 from 1 to get 0. 
k=n 
To rewrite as a function
function
A set of ordered pairs where each first element is paired with one and only one second element and no element in either pair is without a partner.  of n (e.g. f(n)), write the equation
equation
A mathematical statement that says two expressions have the same value; any number sentence with an = .  so that k is by itself on one side of the equal sign and an expression
expression
A mathematical symbol, or combination of symbols, representing a value, or relation. Example: 2+2=4 involving only n is on the other side. 
f(n)=n  
