It is necessary to pass in each subject out of 7 subjects in an examination. In how many ways can a student failed?

Answer

To solve the problem of how many ways a student can fail in at least one of the 7 subjects, we can follow these steps:

### Step 1: Understand the Options for Each Subject
Each subject has two possible outcomes: the student can either pass (P) or fail (F).

### Step 2: Calculate Total Outcomes
Since there are 7 subjects, and each subject has 2 outcomes, the total number of outcomes for all subjects can be calculated using the formula for combinations. The total number of ways to pass or fail in all subjects is given by:
\[ \text{Total Outcomes} = 2^7 \]

### Step 3: Calculate the Outcomes for Passing All Subjects
If the student passes all subjects, there is only 1 way to achieve this outcome (P for each subject). Therefore, the number of ways to pass all subjects is:
\[ \text{Ways to Pass} = 1 \]

### Step 4: Calculate the Ways to Fail in At Least One Subject
To find the number of ways the student can fail in at least one subject, we subtract the number of ways to pass all subjects from the total outcomes:
\[ \text{Ways to Fail} = \text{Total Outcomes} - \text{Ways to Pass} \]
Substituting the values we calculated:
\[ \text{Ways to Fail} = 2^7 - 1 \]

### Step 5: Calculate the Final Answer
Now we compute \(2^7\):
\[ 2^7 = 128 \]
Thus, the number of ways the student can fail in at least one subject is:
\[ \text{Ways to Fail} = 128 - 1 = 127 \]

### Final Answer
The student can fail in 127 different ways.

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