math,combinations,permutation , finding the number of ways to generate a number


finding the number of ways to generate a number

Question:

Tag: math,combinations,permutation

I am new to permutations and combinations. I am provided with the number n and I can make a number with the help of n+1 digits which are 0,1,...n. I need to find in how many ways I can make the sum n by putting these n+1 digits in n places.

Like i am having number n=2
 then 
(a) 0 and 2 (b)1 and 1 (c)2 and 0 . 
for a number n=3
 then 
(a)0,0,3 (b)0,3,0 (c)3,0,0 (d)0,1,2 (e)0,2,1 (f)1,0,2 (g)1,2,0 (h)2,0,1 (i)2,1,0 (j)1,1,1
So  in total i have 10 ways to generate sum=3 by using the digits 0,1,2,3.

And also consider i can put these n+1 digits in n places only.


Answer:

The number of compositions of n into k nonnegative summands is (n+k-1) choose n by the stars-and-bars method. You have k=n, so the count is 2n-1 choose n. Your examples were 3C2=3 and 5C3=10.


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