Fritz is taking an examination that consists of two parts , A and B , with the following instructions:
Part A contains three questions, and a student must answer tow .
Part B contains four questions, and a student must answer two.
Part A must be completed before starting part B.
How many ways can the test be completed?

Options

A. `12`
B. `15`
C. `36`
D. `72`

Answer

To solve the problem step by step, we will calculate the number of ways Fritz can complete the examination by choosing questions from both parts A and B.

### Step 1: Calculate the number of ways to choose questions from Part A
Part A contains 3 questions, and Fritz must answer 2 of them. The number of ways to choose 2 questions from 3 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 from 3} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2! \cdot 1!} = \frac{3 \times 2 \times 1}{(2 \times 1) \cdot (1)} = 3 \]

### Step 2: Calculate the number of ways to choose questions from Part B
Part B contains 4 questions, and Fritz must answer 2 of them. The number of ways to choose 2 questions from 4 can also be calculated using the combination formula:

\[ \text{Number of ways to choose 2 from 4} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1) \cdot (2 \times 1)} = \frac{24}{4} = 6 \]

### Step 3: Calculate the total number of ways to complete the test
Since Part A must be completed before Part B, we multiply the number of ways to choose questions from both parts:

\[ \text{Total ways} = \text{Ways from Part A} \times \text{Ways from Part B} = 3 \times 6 = 18 \]

### Final Answer
Thus, the total number of ways Fritz can complete the test is **18**.