To pass an examinations a students has to pass in each of the 3 papers. In how many ways can a students fail in the examination?

Answer

To solve the problem of how many ways a student can fail in the examination when there are 3 papers, we can follow these steps:

### Step 1: Understand the Choices
Each student has two choices for each paper: they can either pass or fail.

### Step 2: Calculate Total Outcomes
Since there are 3 papers, and each paper has 2 choices (pass or fail), the total number of outcomes can be calculated using the formula for combinations:
\[ \text{Total Outcomes} = 2^n \]
where \( n \) is the number of papers. Here, \( n = 3 \):
\[ \text{Total Outcomes} = 2^3 = 8 \]

### Step 3: Identify Failing Outcomes
Out of these 8 outcomes, we need to find the number of ways in which a student can fail. The student must fail in at least one paper to be considered as failing the examination.

### Step 4: Calculate Failing Outcomes
The only scenario where a student does not fail is when they pass all papers. There is only 1 way for this to happen (pass, pass, pass).

### Step 5: Subtract to Find Failing Outcomes
To find the number of ways a student can fail, we subtract the one outcome where the student passes all papers from the total outcomes:
\[ \text{Failing Outcomes} = \text{Total Outcomes} - \text{Passing Outcome} \]
\[ \text{Failing Outcomes} = 8 - 1 = 7 \]

### Conclusion
Thus, a student can fail in the examination in **7 different ways**.

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