What will be the value of this binomial sum?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
3
down vote

favorite












How to evaluate this binomial sum?
$$sum_r=1^nrn-1 choose r-1(k-1)^r-1$$






combinatorics summation binomial-coefficients







share|cite|improve this question

























    up vote
    3
    down vote

    favorite












    How to evaluate this binomial sum?
    $$sum_r=1^nrn-1 choose r-1(k-1)^r-1$$






    combinatorics summation binomial-coefficients







    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      How to evaluate this binomial sum?
      $$sum_r=1^nrn-1 choose r-1(k-1)^r-1$$






      combinatorics summation binomial-coefficients







      share|cite|improve this question













      How to evaluate this binomial sum?
      $$sum_r=1^nrn-1 choose r-1(k-1)^r-1$$






      combinatorics summation binomial-coefficients




      combinatorics summation binomial-coefficients






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      Debabrata

      1235




      1235




















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          3
          down vote



          accepted










          Hint: let $x=k-1$ begineqnarraysum_r=1^nrn-1 choose r-1x^r-1&=&Big(sum_r=1^nn-1 choose r-1x^r Big)'\
          &=&Big(xunderbracesum_r=1^nn-1 choose r-1x^r-1 Big)'\
          &=&Big(x(1+x)^n-1 Big)'\
          endeqnarray






          share|cite|improve this answer





























            up vote
            1
            down vote













            Hint:



            $$rbinomn-1r-1=binomn-1r-1+(r-1)binomn-1r-1$$



            $$=binomn-1r-1+(n-1)cdotdfrac(n-2)!(r-2)!(n-2)-(r-2)$$



            $$=binomn-1r-1+(n-1)binomn-2r-2$$



            Now put $x=y=1$ in $$(x+y)^m=sum_r=0^mbinom mr x^m-ry^r$$



            Finally put $m=n-1$ and $m=n-2$






            share|cite|improve this answer




















              Your Answer




              StackExchange.ifUsing("editor", function ()
              return StackExchange.using("mathjaxEditing", function ()
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              );
              );
              , "mathjax-editing");

              StackExchange.ready(function()
              var channelOptions =
              tags: "".split(" "),
              id: "69"
              ;
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function()
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled)
              StackExchange.using("snippets", function()
              createEditor();
              );

              else
              createEditor();

              );

              function createEditor()
              StackExchange.prepareEditor(
              heartbeatType: 'answer',
              convertImagesToLinks: true,
              noModals: false,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              );



              );













               

              draft saved


              draft discarded


















              StackExchange.ready(
              function ()
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2965788%2fwhat-will-be-the-value-of-this-binomial-sum%23new-answer', 'question_page');

              );

              Post as a guest

















              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              3
              down vote



              accepted










              Hint: let $x=k-1$ begineqnarraysum_r=1^nrn-1 choose r-1x^r-1&=&Big(sum_r=1^nn-1 choose r-1x^r Big)'\
              &=&Big(xunderbracesum_r=1^nn-1 choose r-1x^r-1 Big)'\
              &=&Big(x(1+x)^n-1 Big)'\
              endeqnarray






              share|cite|improve this answer


























                up vote
                3
                down vote



                accepted










                Hint: let $x=k-1$ begineqnarraysum_r=1^nrn-1 choose r-1x^r-1&=&Big(sum_r=1^nn-1 choose r-1x^r Big)'\
                &=&Big(xunderbracesum_r=1^nn-1 choose r-1x^r-1 Big)'\
                &=&Big(x(1+x)^n-1 Big)'\
                endeqnarray






                share|cite|improve this answer
























                  up vote
                  3
                  down vote



                  accepted







                  up vote
                  3
                  down vote



                  accepted






                  Hint: let $x=k-1$ begineqnarraysum_r=1^nrn-1 choose r-1x^r-1&=&Big(sum_r=1^nn-1 choose r-1x^r Big)'\
                  &=&Big(xunderbracesum_r=1^nn-1 choose r-1x^r-1 Big)'\
                  &=&Big(x(1+x)^n-1 Big)'\
                  endeqnarray






                  share|cite|improve this answer














                  Hint: let $x=k-1$ begineqnarraysum_r=1^nrn-1 choose r-1x^r-1&=&Big(sum_r=1^nn-1 choose r-1x^r Big)'\
                  &=&Big(xunderbracesum_r=1^nn-1 choose r-1x^r-1 Big)'\
                  &=&Big(x(1+x)^n-1 Big)'\
                  endeqnarray







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited 3 hours ago

























                  answered 3 hours ago









                  greedoid

                  32k114287




                  32k114287




















                      up vote
                      1
                      down vote













                      Hint:



                      $$rbinomn-1r-1=binomn-1r-1+(r-1)binomn-1r-1$$



                      $$=binomn-1r-1+(n-1)cdotdfrac(n-2)!(r-2)!(n-2)-(r-2)$$



                      $$=binomn-1r-1+(n-1)binomn-2r-2$$



                      Now put $x=y=1$ in $$(x+y)^m=sum_r=0^mbinom mr x^m-ry^r$$



                      Finally put $m=n-1$ and $m=n-2$






                      share|cite|improve this answer
























                        up vote
                        1
                        down vote













                        Hint:



                        $$rbinomn-1r-1=binomn-1r-1+(r-1)binomn-1r-1$$



                        $$=binomn-1r-1+(n-1)cdotdfrac(n-2)!(r-2)!(n-2)-(r-2)$$



                        $$=binomn-1r-1+(n-1)binomn-2r-2$$



                        Now put $x=y=1$ in $$(x+y)^m=sum_r=0^mbinom mr x^m-ry^r$$



                        Finally put $m=n-1$ and $m=n-2$






                        share|cite|improve this answer






















                          up vote
                          1
                          down vote










                          up vote
                          1
                          down vote









                          Hint:



                          $$rbinomn-1r-1=binomn-1r-1+(r-1)binomn-1r-1$$



                          $$=binomn-1r-1+(n-1)cdotdfrac(n-2)!(r-2)!(n-2)-(r-2)$$



                          $$=binomn-1r-1+(n-1)binomn-2r-2$$



                          Now put $x=y=1$ in $$(x+y)^m=sum_r=0^mbinom mr x^m-ry^r$$



                          Finally put $m=n-1$ and $m=n-2$






                          share|cite|improve this answer












                          Hint:



                          $$rbinomn-1r-1=binomn-1r-1+(r-1)binomn-1r-1$$



                          $$=binomn-1r-1+(n-1)cdotdfrac(n-2)!(r-2)!(n-2)-(r-2)$$



                          $$=binomn-1r-1+(n-1)binomn-2r-2$$



                          Now put $x=y=1$ in $$(x+y)^m=sum_r=0^mbinom mr x^m-ry^r$$



                          Finally put $m=n-1$ and $m=n-2$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 2 hours ago









                          lab bhattacharjee

                          217k14153268




                          217k14153268



























                               

                              draft saved


                              draft discarded















































                               


                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function ()
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2965788%2fwhat-will-be-the-value-of-this-binomial-sum%23new-answer', 'question_page');

                              );

                              Post as a guest
































































                              Comments

                              Post a Comment

                              Popular posts from this blog

                              What does second last employer means? [closed]

                              Image
                              Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote -3 down vote favorite Say, for example I have worked in the following companies, Company A => 2010 - 2011 Company B => 2011 - 2012 Company C => 2012 - Till Date Company D => Have offer Now assume Company D is asking to submit documents from my second last employer. So from the list of companies given, which company would be qualified as second last employer? software-industry employer share | improve this question asked Aug 10 '14 at 5:16 Rajaprabhu Aravindasamy 559 1 6 13 closed as off-topic by Jim G., jmort253 ♦ Aug 11 '14 at 1:58 This question appears to be off-topic. The users who voted to close gave this specific reason: "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address...
                              Read more

                              List of Gilmore Girls characters

                              Image
                              The season three cast—(Back row): Lane, Michel, Paris, Emily, Richard, Sookie, Miss Patty, Kirk; Front row: Jess, Luke, Lorelai, Rory, Dean This is a list of characters for the comedy-drama television series Gilmore Girls . Contents 1 Main characters 1.1 Lorelai Gilmore 1.2 Rory Gilmore 1.3 Sookie St. James 1.4 Lane Kim 1.5 Michel Gerard 1.6 Luke Danes 1.7 Emily Gilmore 1.8 Richard Gilmore 1.9 Paris Geller 1.10 Dean Forester 1.11 Jess Mariano 1.12 Kirk Gleason 1.13 Jason Stiles 1.14 Logan Huntzberger 2 Major recurring characters 2.1 Christopher Hayden 2.2 Jackson Belleville 2.3 Mrs. Kim 2.4 Tristin DuGray 2.5 Max Medina 2.6 Taylor Doose 2.7 Dave Rygalski 2.8 Marty 3 Recurring characters 3.1 Stars Hollow 3.2 Chilton 3.3 Yale 3.4 Other 4 Notable guest stars 5 Unseen characters 6 Notes 7 References Main characters Actor Character Appearances Season 1 Season 2 Season 3 Season 4 Season 5 Season 6 Season 7 A Year in the Life Lauren Graham Lorelai Gilmore...
                              Read more

                              Confectionery

                              Image
                              Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of ...
                              Read more