A student is free to choose only Chemistry, only Biology or both. If out of $32$ students, Chemistry has been chosen by $16$ and Biology by $25$, then how many students have chosen Biology but not Chemistry?
Calculating Students Choosing Biology Only
Problem Analysis
We are given the total number of students and the number of students who chose Chemistry and Biology. We need to find the number of students who selected Biology exclusively.
- Total students ($N$): $32$
- Students who chose Chemistry ($|C|$): $16$
- Students who chose Biology ($|B|$): $25$
- Students could choose only Chemistry, only Biology, or both. This implies that the union of the sets covers all students: $|C \cup B| = 32$.
- We need to find the number of students who chose Biology but not Chemistry ($|B \setminus C|$).
Step-by-Step Solution
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Use the Principle of Inclusion-Exclusion for two sets:
$|C \cup B| = |C| + |B| - |C \cap B|$
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Substitute the given values into the formula to find the number of students who chose both subjects ($|C \cap B|$):
$32 = 16 + 25 - |C \cap B|$
$32 = 41 - |C \cap B|$
$|C \cap B| = 41 - 32$
$|C \cap B| = 9$
So, 9 students chose both Chemistry and Biology.
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Calculate the number of students who chose only Biology. This is the total number of students who chose Biology minus those who chose both Biology and Chemistry:
$|B \setminus C| = |B| - |C \cap B|$
$|B \setminus C| = 25 - 9$
$|B \setminus C| = 16$
Therefore, 16 students have chosen Biology but not Chemistry.
Important Questions from Set Theory and Venn Diagram (Notes)
- Fourteen of the students in a class are girls. Eight students in the class wear blue shirts. Two are neither girls nor wear blue shirts. Five students who wear blue shirts are girls. How many students are there in the class?
- In a group of 44 players, 26 play hockey, 24 play football and 24 play cricket. Eight of them play both hockey and football, 12 play both football and cricket, and 5 play all the three games. How many play both hockey and cricket?
- In a group of students, 30% play only cricket, 20% play only football and 10% play only basketball. 20% of the students play both football and cricket, 15% play both basketball and cricket, 10% play both football and basketball. 15 students play no games, while 5% of the students play all three games. What is the total number of students?
- Which of the following statements are true ?
A. Set A = {x : x $\in$ R and 2 < x < 3} is a null set.
B. Set A = {x : x $\in$ R and 2 < x < 4} is a singleton set.
C. Set A = {x : x $\in$ R and 1 < x < 9} is an infinite set.
D. Set A = {x : x $\in$ R and 1 < x < 9} is a finite set.
E. Set A = {a, b, c, d, e} and B = {c, d, a, e, b} are equal sets.
Choose the correct answer from the options given below : - Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. 21 professors attended Mumbai meeting, 27 attended Delhi meeting and 30 attended Chennai meeting. How many of them attended both the Chennai and Delhi meetings?
