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We know the No. of Non decreasing Sequence of length N is (9+N)CN How can we find the number of decreasing Sequence in a Range [a,b] of length 1 to N;

mvcd
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1 Answers1

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First of all the number of non decreasing sequences of $[a,b]$ of lenght $M$ such that $M\in N$ equals to the number of non decreasing sequences of lenght $M$ with range $[1,a-b+1]$ (to see it you can transform every sequence in the first interval to a sequence in the second interval by subtracting $a-1$ from it).

Now if we take $N=a-b+1$ the question becaomes: what is the number of decreasing sequences of length $M$ in the range $[1,N]$ and the answer is: $$\displaystyle \sum_{r=1}^{M}{N \choose r}{M-1 \choose r-1}$$

Elaqqad
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  • This question is asked again and again. I think we should encourage the people to search for it themselves (or even better, to enable them to do it themselves). You should not give a complete answer. – Dietrich Burde Apr 07 '15 at 14:11
  • Did I need to delete my answer! – Elaqqad Apr 07 '15 at 14:14
  • I did not find any completely duplicate question, But I reduced the problem to a question which is already in this site and the reduction is very simple I don't know what to do here! – Elaqqad Apr 07 '15 at 14:27
  • Yes you caught me, but this question was asked more the $10$ times and always I made it duplicate of this or I linked it to this one, and this is the first time I gave a "second" answer for the same question (you can see the linked questions to the given post) and if you consult my profile you will never find two answers for the same question ever!! – Elaqqad Apr 07 '15 at 14:38