Win_odd Dhamnekar
Junior Member
- Joined
- Aug 14, 2018
- Messages
- 207
Determine the number of vectors \(\displaystyle (x_1,...,x_n)\) such that each \(\displaystyle x_i\) is a nonnegative integer and [math]\displaystyle\sum_{i=0}^n x_i \leq k[/math]
My answer:
Since there are \(\displaystyle \binom{j-1}{n-1} \) nonnegative vectors whose sum is j, there must be \(\displaystyle \displaystyle\sum_{j=n}^k \binom{j-1}{n-1} \)such vectors.
But the \(\displaystyle \binom{j-1}{n-1} \) is the number of subsets of size n from the set of numbers (1,. . ., k) in which j is the largest element in the subset. Consequently, \(\displaystyle \displaystyle\sum_{j=n}^k \binom{j-1}{n-1}\) is just the total number of subsets of size n from a set of size k, showing that the preceding answer is equal to \(\displaystyle \binom{k}{n}\)
Is the above answer correct?
My answer:
Since there are \(\displaystyle \binom{j-1}{n-1} \) nonnegative vectors whose sum is j, there must be \(\displaystyle \displaystyle\sum_{j=n}^k \binom{j-1}{n-1} \)such vectors.
But the \(\displaystyle \binom{j-1}{n-1} \) is the number of subsets of size n from the set of numbers (1,. . ., k) in which j is the largest element in the subset. Consequently, \(\displaystyle \displaystyle\sum_{j=n}^k \binom{j-1}{n-1}\) is just the total number of subsets of size n from a set of size k, showing that the preceding answer is equal to \(\displaystyle \binom{k}{n}\)
Is the above answer correct?