We were discussing how to compare two models. For example, the number of data points must be larger than the number of parameters. if we increase the number of parameters, it will result in goodness of fit and better Chi-square. Consequently the model with higher number of parameters does better.This is articulated as a metric called the mean squared error which is chi-squared divided by degrees of freedom. MSE uses them as numerator and denominator so it represents a trade-off between over-parameterized model and a lower chi-square. A model with a fewer parameters but lower chi-square is preferred.
#codingexercise
Find the number of increasing subsequences of size k in a given array.
For example, if 1 2 3 4, the increasing subsequence of size 2 are 1 2, 1 3, 1 4, 2 3 , 2 4 , 3 4
The recurrence is therefore
dp[i][j] = 1, where i = 1 and 1 <= j <= n
dp[i][j] = sum(dp[i-1][j]), where 1 < i <= k, i <= j <= n and A[m] < A[j] for (i-1) <= m < j

int GetCountIncreasingSubsequences(int A[], int n, int k)
{
    int [,] dp = new int[k, n];
    int  sum = 0;

    for (int i = 0; i < n; i++)
        dp[0, i] = 1;

    for (int size = 1; size < k; size++) {
       for (int i = size; i < n; i++) {
            dp[size, i] = 0;
            for (int j = size - 1; j < i; j++) {
                if (A[j] < A[i])
                    dp[size, i] += dp[size - 1, j];
            }
        }
    }

    for (int i = k - 1; i < n; i++)
        sum += dp[k - 1, i];

   return sum;
}

The above method also memoizes the counts computed for sizes smaller than k so we can easily lookup the results for anything less than k.