(15) A $0, 1$ matrix with 20 rows and several columns satisfies: all columns are distinct and the number of rows where any two columns both have a 1 is at most 2. Find the minimum number of 1's in the matrix when the number of columns is maximized.
(15) For the matrix with the largest number of columns that satisfies the conditions, if in this matrix the number of 1s in a certain column exceeds 3, then retain any 3 of these 1s and change the rest to 0. This will result in a new matrix with the same number of columns that still meets the requirements. If in this new matrix there are still columns where the number of 1s exceeds 3, repeat the above process. Eventually, a matrix with the maximum number of columns and no more than 3 1s in each column will be obtained, and the maximum number of columns in this matrix is $1 + \mathrm{C}_{20}^{1} + \mathrm{C}_{20}^{2} + \mathrm{C}_{20}^{3}$.

On the other hand, construct a matrix that satisfies the requirements as follows: it includes columns without 1s, as well as all distinct columns with only one 1, two 1s, and three 1s. As mentioned above, this matrix has the maximum number of columns, and among all matrices with the maximum number of columns that satisfy the requirements, this matrix has the fewest 1s. The number of columns in this matrix is $1 + \mathrm{C}_{20}^{1} + \mathrm{C}_{20}^{2} + \mathrm{C}_{20}^{3}$, and the number of 1s in this matrix is $\mathrm{C}_{20}^{1} + 2 \mathrm{C}_{20}^{2} + 3 \mathrm{C}_{20}^{3} = 20 + 380 + 3420 = 3820$.