Is there a simple combinatoric interpretation of this identity? [duplicate]

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This question already has an answer here:



  • Combinatorial proof of summation of $sumlimits_k = 0^n n choose k^2= 2n choose n$

    6 answers



I came across an exercise in which we are asked to prove the identity:



$$2nchoose n=sum_k=0^nnchoose k^2$$



The exercise gives the hint:



$$left(1+xright)^2n=left[(1+x)^nright]^2$$



It's not too difficult to use the hint to prove the identity (the expressions in the identity are the coefficients of $x^n$ in the respective expansions of the expressions in the hint, which of course must be the same number), but I was wondering whether there is a neater equivalent-counting interpretation...



It's clear that $2n choose n$ is the number of ways in which we can choose half the elements in a set (where this is possible): how can we interpret $sum_k=0^nnchoose k^2$ equivalently?










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marked as duplicate by Wojowu, N. F. Taussig combinatorics
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Feb 1 at 14:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 4




    $begingroup$
    Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
    $endgroup$
    – Erick Wong
    Feb 1 at 7:12







  • 4




    $begingroup$
    This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
    $endgroup$
    – Mark Bennet
    Feb 1 at 7:13










  • $begingroup$
    @ErickWong Very neat. Turn your comment into an answer and I'll accept it!
    $endgroup$
    – Richard Ambler
    Feb 1 at 7:32















4












$begingroup$



This question already has an answer here:



  • Combinatorial proof of summation of $sumlimits_k = 0^n n choose k^2= 2n choose n$

    6 answers



I came across an exercise in which we are asked to prove the identity:



$$2nchoose n=sum_k=0^nnchoose k^2$$



The exercise gives the hint:



$$left(1+xright)^2n=left[(1+x)^nright]^2$$



It's not too difficult to use the hint to prove the identity (the expressions in the identity are the coefficients of $x^n$ in the respective expansions of the expressions in the hint, which of course must be the same number), but I was wondering whether there is a neater equivalent-counting interpretation...



It's clear that $2n choose n$ is the number of ways in which we can choose half the elements in a set (where this is possible): how can we interpret $sum_k=0^nnchoose k^2$ equivalently?










share|cite|improve this question









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marked as duplicate by Wojowu, N. F. Taussig combinatorics
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Feb 1 at 14:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.













  • 4




    $begingroup$
    Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
    $endgroup$
    – Erick Wong
    Feb 1 at 7:12







  • 4




    $begingroup$
    This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
    $endgroup$
    – Mark Bennet
    Feb 1 at 7:13










  • $begingroup$
    @ErickWong Very neat. Turn your comment into an answer and I'll accept it!
    $endgroup$
    – Richard Ambler
    Feb 1 at 7:32













4












4








4


1



$begingroup$



This question already has an answer here:



  • Combinatorial proof of summation of $sumlimits_k = 0^n n choose k^2= 2n choose n$

    6 answers



I came across an exercise in which we are asked to prove the identity:



$$2nchoose n=sum_k=0^nnchoose k^2$$



The exercise gives the hint:



$$left(1+xright)^2n=left[(1+x)^nright]^2$$



It's not too difficult to use the hint to prove the identity (the expressions in the identity are the coefficients of $x^n$ in the respective expansions of the expressions in the hint, which of course must be the same number), but I was wondering whether there is a neater equivalent-counting interpretation...



It's clear that $2n choose n$ is the number of ways in which we can choose half the elements in a set (where this is possible): how can we interpret $sum_k=0^nnchoose k^2$ equivalently?










share|cite|improve this question









$endgroup$





This question already has an answer here:



  • Combinatorial proof of summation of $sumlimits_k = 0^n n choose k^2= 2n choose n$

    6 answers



I came across an exercise in which we are asked to prove the identity:



$$2nchoose n=sum_k=0^nnchoose k^2$$



The exercise gives the hint:



$$left(1+xright)^2n=left[(1+x)^nright]^2$$



It's not too difficult to use the hint to prove the identity (the expressions in the identity are the coefficients of $x^n$ in the respective expansions of the expressions in the hint, which of course must be the same number), but I was wondering whether there is a neater equivalent-counting interpretation...



It's clear that $2n choose n$ is the number of ways in which we can choose half the elements in a set (where this is possible): how can we interpret $sum_k=0^nnchoose k^2$ equivalently?





This question already has an answer here:



  • Combinatorial proof of summation of $sumlimits_k = 0^n n choose k^2= 2n choose n$

    6 answers







combinatorics binomial-coefficients






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asked Feb 1 at 7:08









Richard AmblerRichard Ambler

1,298515




1,298515




marked as duplicate by Wojowu, N. F. Taussig combinatorics
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Feb 1 at 14:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by Wojowu, N. F. Taussig combinatorics
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Feb 1 at 14:15


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 4




    $begingroup$
    Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
    $endgroup$
    – Erick Wong
    Feb 1 at 7:12







  • 4




    $begingroup$
    This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
    $endgroup$
    – Mark Bennet
    Feb 1 at 7:13










  • $begingroup$
    @ErickWong Very neat. Turn your comment into an answer and I'll accept it!
    $endgroup$
    – Richard Ambler
    Feb 1 at 7:32












  • 4




    $begingroup$
    Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
    $endgroup$
    – Erick Wong
    Feb 1 at 7:12







  • 4




    $begingroup$
    This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
    $endgroup$
    – Mark Bennet
    Feb 1 at 7:13










  • $begingroup$
    @ErickWong Very neat. Turn your comment into an answer and I'll accept it!
    $endgroup$
    – Richard Ambler
    Feb 1 at 7:32







4




4




$begingroup$
Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
$endgroup$
– Erick Wong
Feb 1 at 7:12





$begingroup$
Yes, this can be done very succinctly. Hint: Write $binomnk^2$ as $binomnkbinomnn-k$.
$endgroup$
– Erick Wong
Feb 1 at 7:12





4




4




$begingroup$
This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
$endgroup$
– Mark Bennet
Feb 1 at 7:13




$begingroup$
This must be a duplicate, I think. But in choosing $n$ from $2n$ think about choosing $k$ to include from the first half and $k$ to omit from the second half.
$endgroup$
– Mark Bennet
Feb 1 at 7:13












$begingroup$
@ErickWong Very neat. Turn your comment into an answer and I'll accept it!
$endgroup$
– Richard Ambler
Feb 1 at 7:32




$begingroup$
@ErickWong Very neat. Turn your comment into an answer and I'll accept it!
$endgroup$
– Richard Ambler
Feb 1 at 7:32










3 Answers
3






active

oldest

votes


















6












$begingroup$

Let $A,B$ be disjoint $n$-element sets; the number of $n$-element subsets of $Acup B$ is $binom2nn$. On the other hand, we can get any $n$-element subset of $Acup B$ in the following way: start with the set $A$; pick a number $k$ from $0$ to $n$; throw out $k$ elements of $A$ and replace them with $k$ elements of $B$. In other words, any $n$-element subset of $Acup B$ has the form $(Asetminus X)cup Y$ where $Xsubseteq A$, $Ysubseteq B$, $|X|=|Y|=k$ for some $kin0,dots,n$. The number of different sets we can get in this way is $sum_k=0^nbinom nkbinom nk$.






share|cite|improve this answer









$endgroup$




















    5












    $begingroup$

    This is also a well know interpretation.



    Let's think of a shortest way from $(0,0)$ to $(n,n)$ through lattice points and parallel to $x$-axis or $y$-axis.



    The number of total way is $2n choose n$.



    And each shortest way should pass one and only one of the diagonal points $(0,n), (1,n-1), ldots, (n,0)$.



    The sum of all way is $sum_k=0^nnchoose k^2$.



    Hence, both value are same.






    share|cite|improve this answer









    $endgroup$




















      4












      $begingroup$

      Another interpretation: As Erick Wong stated in the comments,
      $$binomnk^2 = binomnkbinomnn - k$$
      We can interpret
      $$binom2nn$$
      as the number of ways of selecting $n$ people from a group consisting of $n$ men and $n$ women. The expression
      $$binomnkbinomnn - k$$
      counts the number of ways of selecting exactly $k$ men and $n - k$ women. Since $k$ may vary from $0$ to $n$, the RHS also counts the number of ways of selecting $n$ people from $n$ men and $n$ women. Hence,
      $$binom2nn = sum_k = 0^n binomnkbinomnn - k = sum_k = 0^n binomnk^2$$






      share|cite|improve this answer









      $endgroup$



















        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        6












        $begingroup$

        Let $A,B$ be disjoint $n$-element sets; the number of $n$-element subsets of $Acup B$ is $binom2nn$. On the other hand, we can get any $n$-element subset of $Acup B$ in the following way: start with the set $A$; pick a number $k$ from $0$ to $n$; throw out $k$ elements of $A$ and replace them with $k$ elements of $B$. In other words, any $n$-element subset of $Acup B$ has the form $(Asetminus X)cup Y$ where $Xsubseteq A$, $Ysubseteq B$, $|X|=|Y|=k$ for some $kin0,dots,n$. The number of different sets we can get in this way is $sum_k=0^nbinom nkbinom nk$.






        share|cite|improve this answer









        $endgroup$

















          6












          $begingroup$

          Let $A,B$ be disjoint $n$-element sets; the number of $n$-element subsets of $Acup B$ is $binom2nn$. On the other hand, we can get any $n$-element subset of $Acup B$ in the following way: start with the set $A$; pick a number $k$ from $0$ to $n$; throw out $k$ elements of $A$ and replace them with $k$ elements of $B$. In other words, any $n$-element subset of $Acup B$ has the form $(Asetminus X)cup Y$ where $Xsubseteq A$, $Ysubseteq B$, $|X|=|Y|=k$ for some $kin0,dots,n$. The number of different sets we can get in this way is $sum_k=0^nbinom nkbinom nk$.






          share|cite|improve this answer









          $endgroup$















            6












            6








            6





            $begingroup$

            Let $A,B$ be disjoint $n$-element sets; the number of $n$-element subsets of $Acup B$ is $binom2nn$. On the other hand, we can get any $n$-element subset of $Acup B$ in the following way: start with the set $A$; pick a number $k$ from $0$ to $n$; throw out $k$ elements of $A$ and replace them with $k$ elements of $B$. In other words, any $n$-element subset of $Acup B$ has the form $(Asetminus X)cup Y$ where $Xsubseteq A$, $Ysubseteq B$, $|X|=|Y|=k$ for some $kin0,dots,n$. The number of different sets we can get in this way is $sum_k=0^nbinom nkbinom nk$.






            share|cite|improve this answer









            $endgroup$



            Let $A,B$ be disjoint $n$-element sets; the number of $n$-element subsets of $Acup B$ is $binom2nn$. On the other hand, we can get any $n$-element subset of $Acup B$ in the following way: start with the set $A$; pick a number $k$ from $0$ to $n$; throw out $k$ elements of $A$ and replace them with $k$ elements of $B$. In other words, any $n$-element subset of $Acup B$ has the form $(Asetminus X)cup Y$ where $Xsubseteq A$, $Ysubseteq B$, $|X|=|Y|=k$ for some $kin0,dots,n$. The number of different sets we can get in this way is $sum_k=0^nbinom nkbinom nk$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Feb 1 at 12:04









            bofbof

            52.1k558121




            52.1k558121





















                5












                $begingroup$

                This is also a well know interpretation.



                Let's think of a shortest way from $(0,0)$ to $(n,n)$ through lattice points and parallel to $x$-axis or $y$-axis.



                The number of total way is $2n choose n$.



                And each shortest way should pass one and only one of the diagonal points $(0,n), (1,n-1), ldots, (n,0)$.



                The sum of all way is $sum_k=0^nnchoose k^2$.



                Hence, both value are same.






                share|cite|improve this answer









                $endgroup$

















                  5












                  $begingroup$

                  This is also a well know interpretation.



                  Let's think of a shortest way from $(0,0)$ to $(n,n)$ through lattice points and parallel to $x$-axis or $y$-axis.



                  The number of total way is $2n choose n$.



                  And each shortest way should pass one and only one of the diagonal points $(0,n), (1,n-1), ldots, (n,0)$.



                  The sum of all way is $sum_k=0^nnchoose k^2$.



                  Hence, both value are same.






                  share|cite|improve this answer









                  $endgroup$















                    5












                    5








                    5





                    $begingroup$

                    This is also a well know interpretation.



                    Let's think of a shortest way from $(0,0)$ to $(n,n)$ through lattice points and parallel to $x$-axis or $y$-axis.



                    The number of total way is $2n choose n$.



                    And each shortest way should pass one and only one of the diagonal points $(0,n), (1,n-1), ldots, (n,0)$.



                    The sum of all way is $sum_k=0^nnchoose k^2$.



                    Hence, both value are same.






                    share|cite|improve this answer









                    $endgroup$



                    This is also a well know interpretation.



                    Let's think of a shortest way from $(0,0)$ to $(n,n)$ through lattice points and parallel to $x$-axis or $y$-axis.



                    The number of total way is $2n choose n$.



                    And each shortest way should pass one and only one of the diagonal points $(0,n), (1,n-1), ldots, (n,0)$.



                    The sum of all way is $sum_k=0^nnchoose k^2$.



                    Hence, both value are same.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Feb 1 at 8:58









                    Doyun NamDoyun Nam

                    67119




                    67119





















                        4












                        $begingroup$

                        Another interpretation: As Erick Wong stated in the comments,
                        $$binomnk^2 = binomnkbinomnn - k$$
                        We can interpret
                        $$binom2nn$$
                        as the number of ways of selecting $n$ people from a group consisting of $n$ men and $n$ women. The expression
                        $$binomnkbinomnn - k$$
                        counts the number of ways of selecting exactly $k$ men and $n - k$ women. Since $k$ may vary from $0$ to $n$, the RHS also counts the number of ways of selecting $n$ people from $n$ men and $n$ women. Hence,
                        $$binom2nn = sum_k = 0^n binomnkbinomnn - k = sum_k = 0^n binomnk^2$$






                        share|cite|improve this answer









                        $endgroup$

















                          4












                          $begingroup$

                          Another interpretation: As Erick Wong stated in the comments,
                          $$binomnk^2 = binomnkbinomnn - k$$
                          We can interpret
                          $$binom2nn$$
                          as the number of ways of selecting $n$ people from a group consisting of $n$ men and $n$ women. The expression
                          $$binomnkbinomnn - k$$
                          counts the number of ways of selecting exactly $k$ men and $n - k$ women. Since $k$ may vary from $0$ to $n$, the RHS also counts the number of ways of selecting $n$ people from $n$ men and $n$ women. Hence,
                          $$binom2nn = sum_k = 0^n binomnkbinomnn - k = sum_k = 0^n binomnk^2$$






                          share|cite|improve this answer









                          $endgroup$















                            4












                            4








                            4





                            $begingroup$

                            Another interpretation: As Erick Wong stated in the comments,
                            $$binomnk^2 = binomnkbinomnn - k$$
                            We can interpret
                            $$binom2nn$$
                            as the number of ways of selecting $n$ people from a group consisting of $n$ men and $n$ women. The expression
                            $$binomnkbinomnn - k$$
                            counts the number of ways of selecting exactly $k$ men and $n - k$ women. Since $k$ may vary from $0$ to $n$, the RHS also counts the number of ways of selecting $n$ people from $n$ men and $n$ women. Hence,
                            $$binom2nn = sum_k = 0^n binomnkbinomnn - k = sum_k = 0^n binomnk^2$$






                            share|cite|improve this answer









                            $endgroup$



                            Another interpretation: As Erick Wong stated in the comments,
                            $$binomnk^2 = binomnkbinomnn - k$$
                            We can interpret
                            $$binom2nn$$
                            as the number of ways of selecting $n$ people from a group consisting of $n$ men and $n$ women. The expression
                            $$binomnkbinomnn - k$$
                            counts the number of ways of selecting exactly $k$ men and $n - k$ women. Since $k$ may vary from $0$ to $n$, the RHS also counts the number of ways of selecting $n$ people from $n$ men and $n$ women. Hence,
                            $$binom2nn = sum_k = 0^n binomnkbinomnn - k = sum_k = 0^n binomnk^2$$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Feb 1 at 12:03









                            N. F. TaussigN. F. Taussig

                            44.4k93357




                            44.4k93357












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