Step 1: Understanding the Question.
We need to find how much money is left for expenditure after only the PF deduction and the house rent have been paid out of the gross income.
Step 2: Key Formula or Approach.
Start from the gross income, remove the PF deduction, then remove the house rent, and whatever remains is available for the rest of the spending.
Step 3: Detailed Explanation.
The gross income is
\[
5000 + 0.20 \times 5000 = 6000
\]
The PF deduction is
\[
0.10 \times 5000 = 500
\]
so after PF the family has
\[
6000 - 500 = 5500
\]
The house rent, as found earlier, is 22 percent of this net income:
\[
0.22 \times 5500 = 1210
\]
After paying the house rent, the amount left is
\[
5500 - 1210 = 4290
\]
Step 4: Final Answer.
After PF and house rent, Rs. 4,290 is left for the remaining expenditure, so option (a) is correct.
\[ \boxed{Rs.\ 4290} \]