MHB Proving $\sum_{k=0}^{n} \binom{n}{k}^2 =\dfrac{(2n)!}{(n!)^2}$

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The discussion centers on proving the identity $\sum_{k=0}^{n} \binom{n}{k}^2 = \dfrac{(2n)!}{(n!)^2}$. Participants share various approaches to demonstrate this combinatorial identity, highlighting its significance in combinatorics. The proof often involves interpreting the left side as counting paths in a grid or using generating functions. Several users express appreciation for elegant solutions and insights shared throughout the discussion. This identity is a well-known result in combinatorial mathematics, illustrating the relationship between binomial coefficients and factorials.
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prove:
$\sum_{k=0}^{n} \binom{n}{k}^2
=\dfrac{(2n)!}{(n!)^2}$
 
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Albert said:
prove:
$\sum_{k=0}^{n} \binom{n}{k}^2
=\dfrac{(2n)!}{(n!)^2}$

out of 2n objects n objects can be chosen in

$\dfrac{(2n)!}{(n!)^2}$ ways

now let us make 2n objects into 2 groups of n objects each.

for picking n objects from the set we need k objects from 1st set and n-k from 2nd set and n varies from 0 to n so number of ways

$\sum_{k=0}^{n} \binom{n}{k} \binom{n}{n-k}$

the 2 above are same as it shows the number of ways in 2 different ways

so
$\dfrac{(2n)!}{(n!)^2}= \sum_{k=0}^{n} \binom{n}{k} \binom{n}{n-k}$

now as $\binom{n}{k} = \binom{n}{n-k}$ so we get the result
 
kaliprasad said:
out of 2n objects n objects can be chosen in

$\dfrac{(2n)!}{(n!)^2}$ ways

now let us make 2n objects into 2 groups of n objects each.

for picking n objects from the set we need k objects from 1st set and n-k from 2nd set and n varies from 0 to n so number of ways

$\sum_{k=0}^{n} \binom{n}{k} \binom{n}{n-k}$

the 2 above are same as it shows the number of ways in 2 different ways

so
$\dfrac{(2n)!}{(n!)^2}= \sum_{k=0}^{n} \binom{n}{k} \binom{n}{n-k}$

now as $\binom{n}{k} = \binom{n}{n-k}$ so we get the result
nice solution !
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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