Inclusion exclusion principle counting
$begingroup$
50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited Amsterdam and Copenhagen, and 5 visited Berlin and Copenhagen. But only 2 visited all three cities.
Question : How many students visited Copenhagen, but neither Amsterdam nor Berlin?
I have done the graph and I think the answer is 6 but I would like to learn how to compute it.
Thank you!
combinatorics discrete-mathematics inclusion-exclusion
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
$endgroup$
add a comment |
$begingroup$
50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited Amsterdam and Copenhagen, and 5 visited Berlin and Copenhagen. But only 2 visited all three cities.
Question : How many students visited Copenhagen, but neither Amsterdam nor Berlin?
I have done the graph and I think the answer is 6 but I would like to learn how to compute it.
Thank you!
combinatorics discrete-mathematics inclusion-exclusion
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
$endgroup$
add a comment |
$begingroup$
50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited Amsterdam and Copenhagen, and 5 visited Berlin and Copenhagen. But only 2 visited all three cities.
Question : How many students visited Copenhagen, but neither Amsterdam nor Berlin?
I have done the graph and I think the answer is 6 but I would like to learn how to compute it.
Thank you!
combinatorics discrete-mathematics inclusion-exclusion
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
$endgroup$
50 students traveled to Europe, last year. Of these, 12 visited Amsterdam, 13 went to Berlin, and 15 were in Copenhagen. Some visited two cities: 3 visited both Amsterdam and Berlin, 6 visited Amsterdam and Copenhagen, and 5 visited Berlin and Copenhagen. But only 2 visited all three cities.
Question : How many students visited Copenhagen, but neither Amsterdam nor Berlin?
I have done the graph and I think the answer is 6 but I would like to learn how to compute it.
Thank you!
combinatorics discrete-mathematics inclusion-exclusion
combinatorics discrete-mathematics inclusion-exclusion
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
edited Dec 7 '18 at 23:46
N. F. Taussig
43.9k93355
43.9k93355
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
asked Dec 7 '18 at 23:42
Tom1999Tom1999
445
445
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $|A|$, $|B|$, and $|C|$ denote, respectively, the number of students who visited Amsterdam, Berlin, and Copenhagen.
To find the number of students who visited Copenhagen but neither Amsterdam nor Berlin, we must subtract the number of students who have visited both Amsterdam and Copenhagen and the number of students who have visited both Berlin and Copenhagen from the number who have visited Copenhagen. However, if we do that, we will have subtracted those students who have visited all three cities twice, once for each city they visited in addition to Copenhagen. We only want to subtract them once, so we must add them back. Hence, the number of students who visited Copenhagen but neither Amsterdam nor Berlin is
$$|C| - |A cap C| - |B cap C| + |A cap B cap C| = 15 - 6 - 5 + 2 = 6$$
as you found.
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $|A|$, $|B|$, and $|C|$ denote, respectively, the number of students who visited Amsterdam, Berlin, and Copenhagen.
To find the number of students who visited Copenhagen but neither Amsterdam nor Berlin, we must subtract the number of students who have visited both Amsterdam and Copenhagen and the number of students who have visited both Berlin and Copenhagen from the number who have visited Copenhagen. However, if we do that, we will have subtracted those students who have visited all three cities twice, once for each city they visited in addition to Copenhagen. We only want to subtract them once, so we must add them back. Hence, the number of students who visited Copenhagen but neither Amsterdam nor Berlin is
$$|C| - |A cap C| - |B cap C| + |A cap B cap C| = 15 - 6 - 5 + 2 = 6$$
as you found.
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
$begingroup$
Let $|A|$, $|B|$, and $|C|$ denote, respectively, the number of students who visited Amsterdam, Berlin, and Copenhagen.
To find the number of students who visited Copenhagen but neither Amsterdam nor Berlin, we must subtract the number of students who have visited both Amsterdam and Copenhagen and the number of students who have visited both Berlin and Copenhagen from the number who have visited Copenhagen. However, if we do that, we will have subtracted those students who have visited all three cities twice, once for each city they visited in addition to Copenhagen. We only want to subtract them once, so we must add them back. Hence, the number of students who visited Copenhagen but neither Amsterdam nor Berlin is
$$|C| - |A cap C| - |B cap C| + |A cap B cap C| = 15 - 6 - 5 + 2 = 6$$
as you found.
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
add a comment |
$begingroup$
Let $|A|$, $|B|$, and $|C|$ denote, respectively, the number of students who visited Amsterdam, Berlin, and Copenhagen.
To find the number of students who visited Copenhagen but neither Amsterdam nor Berlin, we must subtract the number of students who have visited both Amsterdam and Copenhagen and the number of students who have visited both Berlin and Copenhagen from the number who have visited Copenhagen. However, if we do that, we will have subtracted those students who have visited all three cities twice, once for each city they visited in addition to Copenhagen. We only want to subtract them once, so we must add them back. Hence, the number of students who visited Copenhagen but neither Amsterdam nor Berlin is
$$|C| - |A cap C| - |B cap C| + |A cap B cap C| = 15 - 6 - 5 + 2 = 6$$
as you found.
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
$endgroup$
Let $|A|$, $|B|$, and $|C|$ denote, respectively, the number of students who visited Amsterdam, Berlin, and Copenhagen.
To find the number of students who visited Copenhagen but neither Amsterdam nor Berlin, we must subtract the number of students who have visited both Amsterdam and Copenhagen and the number of students who have visited both Berlin and Copenhagen from the number who have visited Copenhagen. However, if we do that, we will have subtracted those students who have visited all three cities twice, once for each city they visited in addition to Copenhagen. We only want to subtract them once, so we must add them back. Hence, the number of students who visited Copenhagen but neither Amsterdam nor Berlin is
$$|C| - |A cap C| - |B cap C| + |A cap B cap C| = 15 - 6 - 5 + 2 = 6$$
as you found.
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
answered Dec 7 '18 at 23:54
N. F. TaussigN. F. Taussig
43.9k93355
43.9k93355
add a comment |
add a comment |
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