This is a capital budgeting problem where the manager needs to select the optimal combination of projects within the $1,000,000 budget to maximize the total NPV. Since not all projects can be selected due to budget constraints, we need to explore which combination gives the highest return.

Let's first consider the combinations that stay within the $1,000,000 limit.

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Step-by-Step Feasible Combinations:

1. Project A ($500,000) + Project C ($250,000) + Project E ($500,000)

   • Total Cost: $500,000 + $250,000 + $500,000 = $1,000,000
   • Total NPV: $300,000 + $200,000 + $250,000 = $750,000

2. Project A ($500,000) + Project D ($750,000)

   • Total Cost: $500,000 + $750,000 = $1,250,000 (Exceeds budget, not feasible)

3. Project B ($1,000,000)

   • Total Cost: $1,000,000
   • Total NPV: $400,000

4. Project C ($250,000) + Project D ($750,000)

   • Total Cost: $250,000 + $750,000 = $1,000,000
   • Total NPV: $200,000 + $550,000 = $750,000

5. Project A ($500,000) + Project E ($500,000)

   • Total Cost: $500,000 + $500,000 = $1,000,000
   • Total NPV: $300,000 + $250,000 = $550,000

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Comparison of Feasible Combinations:

1. A + C + E: Total NPV = $750,000
2. B: Total NPV = $400,000
3. C + D: Total NPV = $750,000
4. A + E: Total NPV = $550,000

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Optimal Selection:

The two combinations with the highest NPVs are:

• A + C + E (Total NPV = $750,000)
• C + D (Total NPV = $750,000)

Both are equally good in terms of NPV. The manager can choose either based on other qualitative factors (like risk or strategic alignment). If no such factors come into play, either combination is optimal.

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Would you like further details or clarification on any step?
Here are five related questions to deepen your understanding:

1. How is the Net Present Value (NPV) used to make investment decisions?
2. What is the difference between NPV and Internal Rate of Return (IRR)?
3. How can managers handle scenarios where the budget cannot fully fund all projects?
4. What role do opportunity costs play in project selection?
5. How do qualitative factors impact project selection in real-life capital budgeting?

Tip: In project selection problems, always consider constraints first to eliminate infeasible options before comparing profitability metrics like NPV.