Out of 180 students in a class 7 students failed in all the three subjects viz , Physics , Chemistry , Maths. Only 144 students pass in only subject and 21 students and 21 students pass in only tow subjects.
(a) How many students passed in all the three subject ?
(b) How many students passed in at most tow subjects but atleast in one subject ?

Answer

To solve the problem step by step, we will analyze the given information and use set theory concepts.

### Given Information:
- Total students = 180
- Students who failed in all three subjects = 7
- Students who passed in only one subject = 144
- Students who passed in only two subjects = 21

### Step 1: Calculate the total number of students who failed in at least one subject.
To find the total number of students who failed in at least one subject, we can add the students who failed in all three subjects, those who passed in only one subject, and those who passed in only two subjects.

\[ \text{Total failing students} = \text{Students failing in all subjects} + \text{Students passing in only one subject} + \text{Students passing in only two subjects} \]

\[ \text{Total failing students} = 7 + 144 + 21 = 172 \]

### Step 2: Calculate the number of students who passed in all three subjects.
To find the number of students who passed in all three subjects, we subtract the total number of failing students from the total number of students.

\[ \text{Students passing in all subjects} = \text{Total students} - \text{Total failing students} \]

\[ \text{Students passing in all subjects} = 180 - 172 = 8 \]

### Step 3: Calculate the number of students who passed in at most two subjects but at least in one subject.
Students who passed in at most two subjects but at least in one subject include those who passed in only one subject and those who passed in only two subjects. We do not include those who passed in all three subjects.

\[ \text{Students passing in at most two subjects} = \text{Students passing in only one subject} + \text{Students passing in only two subjects} \]

\[ \text{Students passing in at most two subjects} = 144 + 21 = 165 \]

### Final Answers:
(a) The number of students who passed in all three subjects is **8**.

(b) The number of students who passed in at most two subjects but at least in one subject is **165**.

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