To solve this problem, you can use MATLAB to generate the required combinations and fill the matrix ( C ). The goal is to create a matrix ( C ) where each element ( C_{ij} ) represents the sum of all possible products of i elements, with j of them being from the yes vector.
% Define the vectors
yes = [y1, y2, y3, y4];
no = [n1, n2, n3, n4];
% Define the size of the vectors
n_yes = length(yes);
n_no = length(no);
% Initialize the matrix C
C = zeros(n_yes + n_no, n_yes);
% Loop through the number of elements taken (i)
for i = 1:(n_yes + n_no)
% Loop through the number of correct elements (j)
for j = 1:min(i, n_yes)
% Ensure we do not exceed the number of 'no' elements
if (i - j) <= n_no
% Generate all combinations of 'yes' and 'no' elements
yes_combinations = nchoosek(1:n_yes, j);
no_combinations = nchoosek(1:n_no, i - j);
% Initialize the sum for C_ij
sum_C_ij = 0;
% Loop through all combinations of 'yes' elements
for y_comb = 1:size(yes_combinations, 1)
% Loop through all combinations of 'no' elements
for n_comb = 1:size(no_combinations, 1)
% Calculate the product for the current combination
product = prod(yes(yes_combinations(y_comb, :))) * prod(no(no_combinations(n_comb, :)));
% Add the product to the sum for C_ij
sum_C_ij = sum_C_ij + product;
end
end
% Store the sum in the matrix C
C(i, j) = sum_C_ij;
end
end
end
% Display the matrix C
disp('Matrix C:');
disp(C);
