In an examination a minimum is to be secured in ech of 5 subjects for a pass. In how many ways can a student fail?

Answer

To solve the problem of how many ways a student can fail in an examination of 5 subjects, we can follow these steps:

### Step 1: Understand the Problem
We have 5 subjects, and for a student to pass, they must pass in each subject. Therefore, there are two outcomes for each subject: pass or fail.

### Step 2: Calculate Total Outcomes
For each subject, there are 2 possibilities (pass or fail). Since there are 5 subjects, we can calculate the total number of outcomes using the formula for combinations:
\[ \text{Total Outcomes} = 2^5 \]
Calculating this gives:
\[ 2^5 = 32 \]

### Step 3: Identify the Passing Outcome
Among these 32 outcomes, there is only 1 specific outcome where the student passes all subjects. This is the scenario where the student passes in every subject (s1, s2, s3, s4, s5).

### Step 4: Calculate Failing Outcomes
To find the number of ways a student can fail, we subtract the 1 passing outcome from the total outcomes:
\[ \text{Ways to Fail} = \text{Total Outcomes} - \text{Passing Outcome} \]
Substituting the values we have:
\[ \text{Ways to Fail} = 32 - 1 = 31 \]

### Final Answer
Thus, the number of ways a student can fail in the examination of 5 subjects is **31**.

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