4. The eleven distinct numbers listed below are to be packed into bins of size 40 $$\begin{array} { l l l l l l l l l l l } 15 & 22 & 3 & 9 & 23 & x & 5 & 4 & 18 & 20 & 13 \end{array}$$ It is known that \(x\)

• is an integer less than 40
• is the largest number in the list
  1. Explain why it is not possible to pack the numbers into 3 bins of size 40

Given that it is possible to pack the numbers into 4 bins of size 40

determine the range of values for \(x\)

Use the first-fit bin packing algorithm to determine how the numbers can be packed into bins of size 40

Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly. When the first-fit decreasing bin packing algorithm is applied to the list, neither the 15 nor the 13 is placed in the first bin.

Determine the value of \(x\). You must give reasons for your answer.