In a certain examination, 52 candidates offered biology, 60 history, 96 offered mathhematics , 21 offered biology and history, 22 offered both biology and mathematics, 16 mathematics and history. If 7 candidates offer all the three subjects find:(a) how many candidates were there for the examination
number(biology OR history OR mathematics) =
number(biology) + number(history) + number(math) - number(history ANS biology) - number(history AND math) - number(biology AND math) + number(biology AND history AND mathematics)
you are given each of those values, just plug in and evaluate.
Using what you know about the intersections of sets, and using a Venn diagram, you can see that
(52+60+96)-(21+22+16)+(7) = 156
To find the total number of candidates for the examination, we need to add up the number of candidates who offered each subject.
Let's break down the given information:
- Number of candidates who offered biology (B) = 52
- Number of candidates who offered history (H) = 60
- Number of candidates who offered mathematics (M) = 96
- Number of candidates who offered both biology and history (B ∩ H) = 21
- Number of candidates who offered both biology and mathematics (B ∩ M) = 22
- Number of candidates who offered both mathematics and history (M ∩ H) = 16
- Number of candidates who offered all three subjects (B ∩ H ∩ M) = 7
Now, let's calculate the total number of candidates:
Total number of candidates = B + H + M - (B ∩ H ∪ B ∩ M ∪ M ∩ H) + (B ∩ H ∩ M)
Total number of candidates = 52 + 60 + 96 - (21 + 22 + 16) + 7
Total number of candidates = 156 - 59 + 7
Total number of candidates = 104 + 7
Total number of candidates = 111
Therefore, there were 111 candidates for the examination.
To find the number of candidates for the examination, we can use the principle of inclusion-exclusion.
Let's define the following sets:
A = Candidates who offered biology
B = Candidates who offered history
C = Candidates who offered mathematics
From the information given, we know:
|A| = 52 (number of candidates who offered biology)
|B| = 60 (number of candidates who offered history)
|C| = 96 (number of candidates who offered mathematics)
|A ∩ B| = 21 (number of candidates who offered both biology and history)
|A ∩ C| = 22 (number of candidates who offered both biology and mathematics)
|B ∩ C| = 16 (number of candidates who offered both history and mathematics)
|A ∩ B ∩ C| = 7 (number of candidates who offered all three subjects)
Now, we can use the principle of inclusion-exclusion to find the total number of candidates for the examination:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
|A ∪ B ∪ C| = 52 + 60 + 96 - 21 - 22 - 16 + 7
|A ∪ B ∪ C| = 156
Therefore, there were 156 candidates for the examination.