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There exist positive integers n and k such that

\(32 \binom{6}{6} + 31 \binom{7}{6} + 30 \binom{8}{6} + \dots + 3 \binom{35}{6} + 2 \binom{36}{6} + \binom{37}{6} = \binom{n}{k}\)
Enter the ordered pair (n, k)

 Oct 26, 2019
 #1
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The answer is (42,6).

 Oct 26, 2019
 #2
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a=1;b=2; c =a nCr b; if(c==61523748, goto loop, goto next);loop:printa,b;next:a++;if(a<100, goto2, discard=0;a=1;b=b+1;if(b<100, goto2, 0)

 

[n , k] = 39, 8   and  39, (39 - 8) =39, 31 = 61,523,748

 Oct 26, 2019
 #3
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guest #1 your answer is wrong, it is not (42,6)

 Oct 26, 2019
 #4
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wait so what is your answer guest #2

because 61,523,748 is not an ordered pair. and you need parentheses around it too, be sure to clarify what the answer is

 Oct 26, 2019
edited by Guest  Oct 26, 2019
 #5
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What are [39, 8] and [39, 31] ????. 39 nCr 8 =39 nCr 31 =61,523,748 - which is the total sum of your sequence!!!.

Guest Oct 26, 2019
 #6
avatar+23337 
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There exist positive integers n and k such that

\(32 \dbinom{6}{6} + 31 \dbinom{7}{6} + 30 \dbinom{8}{6} + \dots + 3 \dbinom{35}{6} + 2 \dbinom{36}{6} + \dbinom{37}{6} = \dbinom{n}{k}\)
Enter the ordered pair \((n, k)\)

 

Using the hockey stick identity: \(\sum \limits_{i=r}^{n} \dbinom{i}{r} = \dbinom{n+1}{r+1} \qquad \text{for } n,r\in \mathbb{N},\quad n \geq r\)

see: https://en.wikipedia.org/wiki/Hockey-stick_identity

 

\(\begin{array}{|rcll|} \hline &&\mathbf{ 32 \dbinom{6}{6} + 31 \dbinom{7}{6} + 30 \dbinom{8}{6} + \dots + 3 \dbinom{35}{6} + 2 \dbinom{36}{6} + \dbinom{37}{6} } \\\\ &=& \sum \limits_{i=6}^{37} \dbinom{i}{r} +\sum \limits_{i=6}^{36} \dbinom{i}{r} +\sum \limits_{i=6}^{35} \dbinom{i}{r} +\sum \limits_{i=6}^{34} \dbinom{i}{r} +\sum \limits_{i=6}^{33} \dbinom{i}{r} +\sum \limits_{i=6}^{32} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{31} \dbinom{i}{r} +\sum \limits_{i=6}^{30} \dbinom{i}{r} +\sum \limits_{i=6}^{29} \dbinom{i}{r} +\sum \limits_{i=6}^{28} \dbinom{i}{r} +\sum \limits_{i=6}^{27} \dbinom{i}{r} +\sum \limits_{i=6}^{26} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{25} \dbinom{i}{r} +\sum \limits_{i=6}^{24} \dbinom{i}{r} +\sum \limits_{i=6}^{23} \dbinom{i}{r} +\sum \limits_{i=6}^{22} \dbinom{i}{r} +\sum \limits_{i=6}^{21} \dbinom{i}{r} +\sum \limits_{i=6}^{20} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{19} \dbinom{i}{r} +\sum \limits_{i=6}^{18} \dbinom{i}{r} +\sum \limits_{i=6}^{17} \dbinom{i}{r} +\sum \limits_{i=6}^{16} \dbinom{i}{r} +\sum \limits_{i=6}^{15} \dbinom{i}{r} +\sum \limits_{i=6}^{14} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{13} \dbinom{i}{r} +\sum \limits_{i=6}^{12} \dbinom{i}{r} +\sum \limits_{i=6}^{11} \dbinom{i}{r} +\sum \limits_{i=6}^{10} \dbinom{i}{r} +\sum \limits_{i=6}^{9} \dbinom{i}{r} +\sum \limits_{i=6}^{8} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{7} \dbinom{i}{r} +\sum \limits_{i=6}^{6} \dbinom{i}{r} \\ \\ &=& \dbinom{38}{7} +\dbinom{37}{7} +\dbinom{36}{7} +\dbinom{35}{7} +\dbinom{34}{7} +\dbinom{33}{7} \\ && \dbinom{32}{7} +\dbinom{31}{7} +\dbinom{30}{7} +\dbinom{29}{7} +\dbinom{28}{7} +\dbinom{27}{7} \\ && \dbinom{26}{7} +\dbinom{25}{7} +\dbinom{24}{7} +\dbinom{23}{7} +\dbinom{22}{7} +\dbinom{21}{7} \\ && \dbinom{20}{7} +\dbinom{19}{7} +\dbinom{18}{7} +\dbinom{17}{7} +\dbinom{16}{7} +\dbinom{15}{7} \\ && \dbinom{14}{7} +\dbinom{13}{7} +\dbinom{12}{7} +\dbinom{11}{7} +\dbinom{10}{7} +\dbinom{9}{7} \\ && +\dbinom{8}{7} +\dbinom{7}{7} \\\\ &=& \sum \limits_{i=7}^{38} \dbinom{i}{r} \\\\ &=& \mathbf{\dbinom{39}{8}} \\ \hline \end{array}\)

 

laugh

 Oct 28, 2019

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