, 2023

(a) Five identical coins are placed in the cells of a $3 \times 3$ grid so that there is at most one coin in each cell and there is an odd number of coins in each row and each column. (i) Show two examples of how this could be done. (ii) In how many ways can this be done? (3 marks) (b) The numbers 1 to 9 are arranged in the cells of a $3 \times 3$ grid so that every row and every column have an odd sum. (i) Show two examples of how this could be done. (ii) How many such arrangements are possible? You do not need to multiply out your answer, and may write it as a product, such as $2 \times 7 \times 289$ or $3 \times 17$. (7 marks)