if 80 students studying for a sessional examination, 34 students study english, 29 study history while 42 study geography study english and history, 14 study geography and history and 8 study english and geography while 4 students did not study any of the three subjects. (1) how many students study the three subjects? (2) how many students study any combination of two subjects only?
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To solve this problem, we can use the principle of inclusion-exclusion.
1) To find the number of students studying all three subjects, we subtract the number of students who did not study any of the three subjects from the number of students studying geography, history, and English together.
Number of students studying all three subjects = Number of students studying geography, history, and English together - Number of students who did not study any of the three subjects.
From the given information:
Number of students studying geography, history, and English together = 4
Number of students who did not study any of the three subjects = 4
Therefore, the number of students studying all three subjects = 4 - 4 = 0.
Hence, there are no students who study all three subjects.
2) To find the number of students studying any combination of two subjects only, we need to add up the number of students studying each subject individually and then subtract the number of students studying all three subjects.
Number of students studying any combination of two subjects only = Number of students studying English only + Number of students studying history only + Number of students studying geography only - Number of students studying all three subjects.
From the given information:
Number of students studying English only = Number of students studying English - Number of students studying geography and English - Number of students studying English and history + Number of students studying geography, history, and English
= 34 - 8 - 42 + 0
= -16 (which does not make sense in this context, as we cannot have a negative number of students)
Number of students studying history only = Number of students studying history - Number of students studying geography and history - Number of students studying English and history + Number of students studying geography, history, and English
= 29 - 14 - 42 + 0
= -27 (which does not make sense in this context)
Number of students studying geography only = Number of students studying geography - Number of students studying geography and English - Number of students studying geography and history + Number of students studying geography, history, and English
= Unknown, as the number of students studying geography alone is not given.
Since we do not have the number of students studying geography alone, we cannot find the number of students studying any combination of two subjects only.
Therefore, for part (2), we cannot determine the number of students studying any combination of two subjects only with the given information.