Question

How many nondecreasing sequences of length 10 can you make from the set {0,1,2,...,100}? An example is 0,0,5,10,10,10,15,25,35,92.

Answer 1

Let i = 1,2,3...10 and let each of i appear n_i number of times in a non decreasing sequence of length 10. As the sequence should be non decreasing it should look like this: {...n₁ number of 1's.....n₂ number of 2's..........n₁₀₀number of 100's...}
with subject to constraint: n₁ + n₂ + n₃ +.....+ n₁₀₀ = 10
n₁ + n₂ + n₃ +.....+ n₁₀₀ = 10
The number of non-negative solutions to the above equation gives the total number of non-decreasing sequences. Each solution gives a unique sequence.
Number of non-negative solution :

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