Non-Strictly Increasing Sequences

[Puzzle Prime]

What is the number of non-strictly increasing sequences of M integer numbers between 1 and N?

[Vivek]

Let xi be the number of times the number “i” appears in the sequence.

So, We need to find the number of solutions for the equation

x1+x2+x3+…+xN = M where xi>=0

And That is ((M+N-1) choose (N-1)).

[Puzzle Prime]

That’s true, but how do you calculate the number of solutions of x1+x2+… +xN=M?

[Vivek]

Usually the standard explanation goes like this :- Consider for example , X1+X2+X3=5. The solution can be considered equivalent to 5 *s separated by 2 |s . For example (1,2,2) corresponds to *|**|** . In the general case, M *s and N-1 |s and that gives ((M+N-1) choose (N-1))

[Puzzle Prime]

That’s right:) Alternatively (or more like equivalently), you can add 1 to the second smallest element, 2 to the third smallest element, and so on, and reduce the problem to choosing M different numbers from 1 to N + M – 1.