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Combinatorics: How many persons like apples and pears and strawberries?
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Out of $32$ persons, every person likes to eat at least one of the following type of fruits: Strawberries, Apples and Pears. (Which means that there does not exist any person, who does not like to eat any type of fruit). Furthermore, we know that $20$ persons like to eat apples, $18$ persons like to eat pears and $28$ persons like to eat strawberries.
(a) There are $10$ persons who like apples and pears, $16$ persons who like apples and strawberries, and $12$ persons who like pears and strawberries.
How can I find out how many people like apples as well as pears as well as strawberries?
To structure this a bit:
$32$ persons
$20 rightarrow$ apples ($rightarrow$ meaning "like")
$18 rightarrow$ pears
$28 rightarrow $ strawberries
And for (a)
$10$ persons $rightarrow$ (apples & pears)
$16$ persons $rightarrow$ (apples & strawberries)
$12$ persons $rightarrow$ (pears & strawberries)
Since we know that the total number of persons is $32$.
Can I just do the following?
Because $20$ persons like apples I can just add the following numbers together:
$10 rightarrow$ ($10$ apples & $0$ pears) + $16 rightarrow$ ($6$ apples & $10$ strawberries) $+ 12 rightarrow$ ($12$ pears & $0$ strawberries). So in total I'd get $10 + 16 + 12 = 28$ people who like apples, pears and strawberries? Is that correct?
(b) Assume that you don't have the information in (a). Give the preferably limits for the amount of persons who like to eat all kind of fruits.
Since $18$ person like pears, can I just say that $18$ persons like to eat pears, apples and strawberries? (As $18$ is the minimal amount of fruits).
elementary-set-theory
edited 24 mins ago
Parcly Taxel
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
add a comment |Â
up vote
6
down vote
favorite
Out of $32$ persons, every person likes to eat at least one of the following type of fruits: Strawberries, Apples and Pears. (Which means that there does not exist any person, who does not like to eat any type of fruit). Furthermore, we know that $20$ persons like to eat apples, $18$ persons like to eat pears and $28$ persons like to eat strawberries.
(a) There are $10$ persons who like apples and pears, $16$ persons who like apples and strawberries, and $12$ persons who like pears and strawberries.
How can I find out how many people like apples as well as pears as well as strawberries?
To structure this a bit:
$32$ persons
$20 rightarrow$ apples ($rightarrow$ meaning "like")
$18 rightarrow$ pears
$28 rightarrow $ strawberries
And for (a)
$10$ persons $rightarrow$ (apples & pears)
$16$ persons $rightarrow$ (apples & strawberries)
$12$ persons $rightarrow$ (pears & strawberries)
Since we know that the total number of persons is $32$.
Can I just do the following?
Because $20$ persons like apples I can just add the following numbers together:
$10 rightarrow$ ($10$ apples & $0$ pears) + $16 rightarrow$ ($6$ apples & $10$ strawberries) $+ 12 rightarrow$ ($12$ pears & $0$ strawberries). So in total I'd get $10 + 16 + 12 = 28$ people who like apples, pears and strawberries? Is that correct?
(b) Assume that you don't have the information in (a). Give the preferably limits for the amount of persons who like to eat all kind of fruits.
Since $18$ person like pears, can I just say that $18$ persons like to eat pears, apples and strawberries? (As $18$ is the minimal amount of fruits).
elementary-set-theory
edited 24 mins ago
Parcly Taxel
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Out of $32$ persons, every person likes to eat at least one of the following type of fruits: Strawberries, Apples and Pears. (Which means that there does not exist any person, who does not like to eat any type of fruit). Furthermore, we know that $20$ persons like to eat apples, $18$ persons like to eat pears and $28$ persons like to eat strawberries.
(a) There are $10$ persons who like apples and pears, $16$ persons who like apples and strawberries, and $12$ persons who like pears and strawberries.
How can I find out how many people like apples as well as pears as well as strawberries?
To structure this a bit:
$32$ persons
$20 rightarrow$ apples ($rightarrow$ meaning "like")
$18 rightarrow$ pears
$28 rightarrow $ strawberries
And for (a)
$10$ persons $rightarrow$ (apples & pears)
$16$ persons $rightarrow$ (apples & strawberries)
$12$ persons $rightarrow$ (pears & strawberries)
Since we know that the total number of persons is $32$.
Can I just do the following?
Because $20$ persons like apples I can just add the following numbers together:
$10 rightarrow$ ($10$ apples & $0$ pears) + $16 rightarrow$ ($6$ apples & $10$ strawberries) $+ 12 rightarrow$ ($12$ pears & $0$ strawberries). So in total I'd get $10 + 16 + 12 = 28$ people who like apples, pears and strawberries? Is that correct?
(b) Assume that you don't have the information in (a). Give the preferably limits for the amount of persons who like to eat all kind of fruits.
Since $18$ person like pears, can I just say that $18$ persons like to eat pears, apples and strawberries? (As $18$ is the minimal amount of fruits).
elementary-set-theory
edited 24 mins ago
Parcly Taxel
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
Out of $32$ persons, every person likes to eat at least one of the following type of fruits: Strawberries, Apples and Pears. (Which means that there does not exist any person, who does not like to eat any type of fruit). Furthermore, we know that $20$ persons like to eat apples, $18$ persons like to eat pears and $28$ persons like to eat strawberries.
(a) There are $10$ persons who like apples and pears, $16$ persons who like apples and strawberries, and $12$ persons who like pears and strawberries.
How can I find out how many people like apples as well as pears as well as strawberries?
To structure this a bit:
$32$ persons
$20 rightarrow$ apples ($rightarrow$ meaning "like")
$18 rightarrow$ pears
$28 rightarrow $ strawberries
And for (a)
$10$ persons $rightarrow$ (apples & pears)
$16$ persons $rightarrow$ (apples & strawberries)
$12$ persons $rightarrow$ (pears & strawberries)
Since we know that the total number of persons is $32$.
Can I just do the following?
Because $20$ persons like apples I can just add the following numbers together:
$10 rightarrow$ ($10$ apples & $0$ pears) + $16 rightarrow$ ($6$ apples & $10$ strawberries) $+ 12 rightarrow$ ($12$ pears & $0$ strawberries). So in total I'd get $10 + 16 + 12 = 28$ people who like apples, pears and strawberries? Is that correct?
(b) Assume that you don't have the information in (a). Give the preferably limits for the amount of persons who like to eat all kind of fruits.
Since $18$ person like pears, can I just say that $18$ persons like to eat pears, apples and strawberries? (As $18$ is the minimal amount of fruits).
elementary-set-theory
elementary-set-theory
edited 24 mins ago
Parcly Taxel
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
edited 24 mins ago
Parcly Taxel
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
edited 24 mins ago
Parcly Taxel
38.7k137097
edited 24 mins ago
Parcly Taxel
38.7k137097
edited 24 mins ago
Parcly Taxel
38.7k137097
38.7k137097
asked 1 hour ago
JavaTeachMe2018
37917
asked 1 hour ago
JavaTeachMe2018
37917
asked 1 hour ago
JavaTeachMe2018
37917
37917
Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago
add a comment |Â
Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago
Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago
Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Suppose we add up all the number of people who like to eat one of the three fruits separately. We double-count those liking two and triple-count those liking three. Now if we subtract the people who like to eat two separately, we correctly count those liking two, but fail to count those liking all three. Thus the difference between the count now and the total number of people must give the number of people liking all three fruits:
$$32-(20+18+28-10-16-12)=32-28=4$$
Now suppose we don't have the counts of people liking exactly two fruits. Since 18 people like pears, the maximum number of people liking all three fruits is 18, but we can't put exactly 18 since then 4 people must like only apples and 12 people only strawberries, at which point the number of people is greater than the fixed 32. We can have 17 people liking all three fruits, though, in which case
- 3, 1, 11 like only apples, pears, strawberries respectively
- nobody likes exactly two fruits
For the other extreme, consider the least amount of people who must like at least two fixed fruits:
- 20 like apples and 18 like pears, so since there are 32 people there must be at least $20+18-32=6$ people liking both apples and pears
- Similarly, at least $20+28-32=16$ like apples and strawberries; at least 14, pears and strawberries
Now place these "forced" people in such a way that nobody likes all three fruits. We find that there are two more people than stipulated who like apples, so at least two people like all three fruits, and we get the same result for the other fruits. Thus we must merge six people into two in the centre where people like all three fruits; fortituously the total number of people becomes exactly 32. Thus the minimum number of people who like all three fruits is 2, with
- 14 liking only apples/strawberries
- 12 liking only pears/strawberries
- 4 liking only apples/pears
- nobody likes exactly one fruit.
answered 30 mins ago
Parcly Taxel
38.7k137097
add a comment |Â
up vote
1
down vote
What you have actually worked out for part a) is the number of people who don't like all three. So since there are $32$ people, and $28$ don't like all three, the number of people who like Apples, Pears and Strawberries is $4$.
answered 42 mins ago
MRobinson
1,716219
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Suppose we add up all the number of people who like to eat one of the three fruits separately. We double-count those liking two and triple-count those liking three. Now if we subtract the people who like to eat two separately, we correctly count those liking two, but fail to count those liking all three. Thus the difference between the count now and the total number of people must give the number of people liking all three fruits:
$$32-(20+18+28-10-16-12)=32-28=4$$
Now suppose we don't have the counts of people liking exactly two fruits. Since 18 people like pears, the maximum number of people liking all three fruits is 18, but we can't put exactly 18 since then 4 people must like only apples and 12 people only strawberries, at which point the number of people is greater than the fixed 32. We can have 17 people liking all three fruits, though, in which case
- 3, 1, 11 like only apples, pears, strawberries respectively
- nobody likes exactly two fruits
For the other extreme, consider the least amount of people who must like at least two fixed fruits:
- 20 like apples and 18 like pears, so since there are 32 people there must be at least $20+18-32=6$ people liking both apples and pears
- Similarly, at least $20+28-32=16$ like apples and strawberries; at least 14, pears and strawberries
Now place these "forced" people in such a way that nobody likes all three fruits. We find that there are two more people than stipulated who like apples, so at least two people like all three fruits, and we get the same result for the other fruits. Thus we must merge six people into two in the centre where people like all three fruits; fortituously the total number of people becomes exactly 32. Thus the minimum number of people who like all three fruits is 2, with
- 14 liking only apples/strawberries
- 12 liking only pears/strawberries
- 4 liking only apples/pears
- nobody likes exactly one fruit.
answered 30 mins ago
Parcly Taxel
38.7k137097
add a comment |Â
up vote
2
down vote
accepted
Suppose we add up all the number of people who like to eat one of the three fruits separately. We double-count those liking two and triple-count those liking three. Now if we subtract the people who like to eat two separately, we correctly count those liking two, but fail to count those liking all three. Thus the difference between the count now and the total number of people must give the number of people liking all three fruits:
$$32-(20+18+28-10-16-12)=32-28=4$$
Now suppose we don't have the counts of people liking exactly two fruits. Since 18 people like pears, the maximum number of people liking all three fruits is 18, but we can't put exactly 18 since then 4 people must like only apples and 12 people only strawberries, at which point the number of people is greater than the fixed 32. We can have 17 people liking all three fruits, though, in which case
- 3, 1, 11 like only apples, pears, strawberries respectively
- nobody likes exactly two fruits
For the other extreme, consider the least amount of people who must like at least two fixed fruits:
- 20 like apples and 18 like pears, so since there are 32 people there must be at least $20+18-32=6$ people liking both apples and pears
- Similarly, at least $20+28-32=16$ like apples and strawberries; at least 14, pears and strawberries
Now place these "forced" people in such a way that nobody likes all three fruits. We find that there are two more people than stipulated who like apples, so at least two people like all three fruits, and we get the same result for the other fruits. Thus we must merge six people into two in the centre where people like all three fruits; fortituously the total number of people becomes exactly 32. Thus the minimum number of people who like all three fruits is 2, with
- 14 liking only apples/strawberries
- 12 liking only pears/strawberries
- 4 liking only apples/pears
- nobody likes exactly one fruit.
answered 30 mins ago
Parcly Taxel
38.7k137097
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Suppose we add up all the number of people who like to eat one of the three fruits separately. We double-count those liking two and triple-count those liking three. Now if we subtract the people who like to eat two separately, we correctly count those liking two, but fail to count those liking all three. Thus the difference between the count now and the total number of people must give the number of people liking all three fruits:
$$32-(20+18+28-10-16-12)=32-28=4$$
Now suppose we don't have the counts of people liking exactly two fruits. Since 18 people like pears, the maximum number of people liking all three fruits is 18, but we can't put exactly 18 since then 4 people must like only apples and 12 people only strawberries, at which point the number of people is greater than the fixed 32. We can have 17 people liking all three fruits, though, in which case
- 3, 1, 11 like only apples, pears, strawberries respectively
- nobody likes exactly two fruits
For the other extreme, consider the least amount of people who must like at least two fixed fruits:
- 20 like apples and 18 like pears, so since there are 32 people there must be at least $20+18-32=6$ people liking both apples and pears
- Similarly, at least $20+28-32=16$ like apples and strawberries; at least 14, pears and strawberries
Now place these "forced" people in such a way that nobody likes all three fruits. We find that there are two more people than stipulated who like apples, so at least two people like all three fruits, and we get the same result for the other fruits. Thus we must merge six people into two in the centre where people like all three fruits; fortituously the total number of people becomes exactly 32. Thus the minimum number of people who like all three fruits is 2, with
- 14 liking only apples/strawberries
- 12 liking only pears/strawberries
- 4 liking only apples/pears
- nobody likes exactly one fruit.
answered 30 mins ago
Parcly Taxel
38.7k137097
Suppose we add up all the number of people who like to eat one of the three fruits separately. We double-count those liking two and triple-count those liking three. Now if we subtract the people who like to eat two separately, we correctly count those liking two, but fail to count those liking all three. Thus the difference between the count now and the total number of people must give the number of people liking all three fruits:
$$32-(20+18+28-10-16-12)=32-28=4$$
Now suppose we don't have the counts of people liking exactly two fruits. Since 18 people like pears, the maximum number of people liking all three fruits is 18, but we can't put exactly 18 since then 4 people must like only apples and 12 people only strawberries, at which point the number of people is greater than the fixed 32. We can have 17 people liking all three fruits, though, in which case
- 3, 1, 11 like only apples, pears, strawberries respectively
- nobody likes exactly two fruits
For the other extreme, consider the least amount of people who must like at least two fixed fruits:
- 20 like apples and 18 like pears, so since there are 32 people there must be at least $20+18-32=6$ people liking both apples and pears
- Similarly, at least $20+28-32=16$ like apples and strawberries; at least 14, pears and strawberries
Now place these "forced" people in such a way that nobody likes all three fruits. We find that there are two more people than stipulated who like apples, so at least two people like all three fruits, and we get the same result for the other fruits. Thus we must merge six people into two in the centre where people like all three fruits; fortituously the total number of people becomes exactly 32. Thus the minimum number of people who like all three fruits is 2, with
- 14 liking only apples/strawberries
- 12 liking only pears/strawberries
- 4 liking only apples/pears
- nobody likes exactly one fruit.
answered 30 mins ago
Parcly Taxel
38.7k137097
answered 30 mins ago
Parcly Taxel
38.7k137097
answered 30 mins ago
Parcly Taxel
38.7k137097
answered 30 mins ago
Parcly Taxel
38.7k137097
38.7k137097
add a comment |Â
add a comment |Â
up vote
1
down vote
What you have actually worked out for part a) is the number of people who don't like all three. So since there are $32$ people, and $28$ don't like all three, the number of people who like Apples, Pears and Strawberries is $4$.
answered 42 mins ago
MRobinson
1,716219
add a comment |Â
up vote
1
down vote
What you have actually worked out for part a) is the number of people who don't like all three. So since there are $32$ people, and $28$ don't like all three, the number of people who like Apples, Pears and Strawberries is $4$.
answered 42 mins ago
MRobinson
1,716219
add a comment |Â
up vote
1
down vote
up vote
1
down vote
What you have actually worked out for part a) is the number of people who don't like all three. So since there are $32$ people, and $28$ don't like all three, the number of people who like Apples, Pears and Strawberries is $4$.
answered 42 mins ago
MRobinson
1,716219
What you have actually worked out for part a) is the number of people who don't like all three. So since there are $32$ people, and $28$ don't like all three, the number of people who like Apples, Pears and Strawberries is $4$.
answered 42 mins ago
MRobinson
1,716219
edited 25 mins ago
answered 42 mins ago
MRobinson
1,716219
answered 42 mins ago
MRobinson
1,716219
answered 42 mins ago
MRobinson
1,716219
1,716219
add a comment |Â
add a comment |Â
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Your answer to a) cannot be right, because only 10 persons like apples and pears
– Patricio
53 mins ago