Question

A teacher prepares a test with 5 objective questions, of which 4 have to be answered. The first 2 questions each have 3 choices and the last 3 each have 4 choices. Find the total number of ways to answer the paper.

• A. 255
• B. 816
• C. 192
• D. 100
• E. 144

Hint:

“Exactly \(k\) of \(n\) questions” type problems mein pehle choose karo kaun se attempt honge (ya kis ko chhoda jayega), phir har chosen question ke liye choices multiply kar do (Multiplication Principle).

Answer 1

The Correct Option is B

Step 1: Understand the structure.
There are 5 questions \(Q_1, Q_2, Q_3, Q_4, Q_5\). Exactly 4 must be answered.
Choices per question: \(Q_1, Q_2\) have \(3\) choices each; \(Q_3, Q_4, Q_5\) have \(4\) choices each.

Step 2: Casework by which question is left unanswered.
\(\bullet\) Leave out \(Q_1\): Answer \(Q_2(C)\), \(Q_3(D)\), \(Q_4(D)\), \(Q_5(D)\) \(\Rightarrow 3 \times 4 \times 4 \times 4 = 192\).
\(\bullet\) Leave out \(Q_2\): Similarly \(3 \times 4 \times 4 \times 4 = 192\).
\(\bullet\) Leave out \(Q_3\): Answer \(Q_1(C)\), \(Q_2(C)\), \(Q_4(D)\), \(Q_5(D)\) \(\Rightarrow 3 \times 3 \times 4 \times 4 = 144\).
\(\bullet\) Leave out \(Q_4\): Again \(3 \times 3 \times 4 \times 4 = 144\).
\(\bullet\) Leave out \(Q_5\): Again \(3 \times 3 \times 4 \times 4 = 144\).

Step 3: Add the cases (Addition Principle).
Total \(= 192 + 192 + 144 + 144 + 144 = 816\).

Final Answer:
\[ \boxed{816} \]