Equality of three probability formulas with binomial coefficients

Question

I need to argue why the following three formulas are equal:

$$\binom{n+1}{k}(n-k+1) \\ \binom{n}{k}(n+1) \\ \binom{n+1}{k+1}(k+1) $$

I've already rewritten the formulas so they equal, but I do not know how to explain why they are equal in words. The formulas describe a lottery game where k lotto numbers are drawn from n+1 numbers. Another lotto number is drawn as an additional number.

Basically, I know what the binomial coefficient does, but I do not know what the multiplication has to do with it.

Could somebody please help me here?

Thanks

Answer 1

I'd think a little different. Suppose we have $n+1$ people, and you want to choose a chaired committee of $k+1$ members with a chairperson among the committee members. The 3 formulas correspond to 3 ways to do it:

• You can choose first $k$ members of committee and then choose a chairperson;
• You can choose first a chairperson and then choose $k$ members of committee;
• You can choose first $k+1$ members of committee and then choose a chairperson among the committee members;

Answer 2

Hint: $\ {a\choose b}=\frac{a!}{b!(a-b)!}\ $ and $ (a+1)! = a!(a+1)\ $. Instances of these identities are all you need to use prove that your three expressions are equal.