Let n and k be integers with 1 ≤k ≤n. Prove that ∑k=1^n( n k )( n k-1 )=(1)/

Let n and k be integers with 1 ≤k ≤n. Prove that ∑k=1^n( ...

Let $n$ and $k$ be integers with $1 \leq k \leq n$. Prove that $$ \sum_{k=1}^{n}\left(\begin{array}{l} n \\ k \end{array}\right)\left(\begin{array}{c} n \\ k-1 \end{array}\right)=\frac{1}{2}\left(\begin{array}{c} 2 n+1 \\ n+1 \end{array}\right)-\left(\begin{array}{c} 2 n \\ n \end{array}\right) $$.

Let $n$ and $k$ be integers with $1 \leq k \leq n$. Prove that
$$
\sum_{k=1}^{n}\left(\begin{array}{l}
n \\
k
\end{array}\right)\left(\begin{array}{c}
n \\
k-1
\end{array}\right)=\frac{1}{2}\left(\begin{array}{c}
2 n+1 \\
n+1
\end{array}\right)-\left(\begin{array}{c}
2 n \\
n
\end{array}\right)
$$.

Key Concepts

Binomial Coefficients

These are the basic building blocks in combinatorics; they count the number of ways to choose a subset of items from a larger set. Their algebraic properties, such as symmetry and recurrence relations, are frequently used to establish more complex identities and to interpret combinatorial sums.

Combinatorial Identities

These are equalities involving combinatorial quantities (often binomial coefficients) that hold for all natural numbers involved. They are proved by algebraic manipulation, generating functions, or by combinatorial arguments like double counting, which involves counting the same quantity in different ways.

Vandermonde’s Identity

A specific combinatorial identity that expresses the sum of products of binomial coefficients in a closed form. It is central to many proofs involving sums over binomial products and is used to simplify and interpret sums where the indices of the binomial coefficients vary in a coordinated way.

Generating Functions

These are series whose coefficients encode sequences such as binomial coefficients. They transform combinatorial problems into problems of algebraic manipulation, making it easier to derive relationships and identities that relate different binomial sums.

Double Counting / Combinatorial Proof

This technique involves counting the same combinatorial object in two different ways to establish an equality. It is a powerful tool in combinatorics for proving identities by providing a conceptual explanation that the algebraic expressions on both sides of the identity actually count the same set of objects.

Transcript

00:01 Okay, so we are going to calculate the term x to the n plus 1 from the polynomial 1 plus x to the 2n in two different ways.

00:11 The first one will be using the binomial theorem that tell us that that term is going to have a coefficient combinatorial number of 2nn plus 1.

00:24 Now, i want you to ignore this for a little bit because it's a little bit unnecessary.

00:34 Okay, so the term, the combinatorial number 2n plus 1 corresponds to 2n factorial over in plus 1 factorial 2n minus n factorial.

00:48 I use an extra step here that it wasn't necessary so you can ignore it.

00:54 Now, that if we simplify this expression, is 2n factorial over n plus 1 factorial and minus 1 factorial.

01:08 Now we can multiply on the top and on the bottom by n.

01:14 And here we can put this together.

01:20 So we put it together.

01:21 And now we have this expression.

01:25 Now n can be written as 2n plus 1 minus n minus 1.

01:35 And so we can separate this into 2n plus 1, all the positives minus n plus 1, all the negatives.

01:43 Now we can break that into two different fractions.

01:48 2n factorial times 2n plus 1 over n plus 1 n factorial.

01:53 And on the other side, 2n factorial, n plus 1 over n plus 1 factorial, n plus 1 factorial, n plus 1 factorial...

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