How many integer solutions with negative numbers?
If a question asked: How many integer solutions of $x_1+x_2+x_3+x_4=30$ with $-9 leq x_i leq 21$? How would this be solved?
I understand how to solve if the inequality was $0 leq x_i leq 21$?, but how to solve between a negative and positive inequality?
$N = binom{30 + 4-1}{30}$
$N(A_i) = binom{(30 - ?) +4-1}{30 - ?}$
...
combinatorics
asked Dec 10 '18 at 20:41
Math Newbie
428
add a comment |
If a question asked: How many integer solutions of $x_1+x_2+x_3+x_4=30$ with $-9 leq x_i leq 21$? How would this be solved?
I understand how to solve if the inequality was $0 leq x_i leq 21$?, but how to solve between a negative and positive inequality?
$N = binom{30 + 4-1}{30}$
$N(A_i) = binom{(30 - ?) +4-1}{30 - ?}$
...
combinatorics
asked Dec 10 '18 at 20:41
Math Newbie
428
It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34
add a comment |
If a question asked: How many integer solutions of $x_1+x_2+x_3+x_4=30$ with $-9 leq x_i leq 21$? How would this be solved?
I understand how to solve if the inequality was $0 leq x_i leq 21$?, but how to solve between a negative and positive inequality?
$N = binom{30 + 4-1}{30}$
$N(A_i) = binom{(30 - ?) +4-1}{30 - ?}$
...
combinatorics
asked Dec 10 '18 at 20:41
Math Newbie
428
If a question asked: How many integer solutions of $x_1+x_2+x_3+x_4=30$ with $-9 leq x_i leq 21$? How would this be solved?
I understand how to solve if the inequality was $0 leq x_i leq 21$?, but how to solve between a negative and positive inequality?
$N = binom{30 + 4-1}{30}$
$N(A_i) = binom{(30 - ?) +4-1}{30 - ?}$
...
combinatorics
combinatorics
asked Dec 10 '18 at 20:41
Math Newbie
428
asked Dec 10 '18 at 20:41
Math Newbie
428
edited Dec 10 '18 at 22:34
asked Dec 10 '18 at 20:41
Math Newbie
428
asked Dec 10 '18 at 20:41
Math Newbie
428
asked Dec 10 '18 at 20:41
Math Newbie
428
428
It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34
add a comment |
It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34
It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34
add a comment |
1 Answer
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Let $y_i=x_i+9$. Then
$$
y_1+y_2+y_3+y_4=66\
0leq y_ileq 30
$$
Now find the number of solutions the way you say you know.
answered Dec 10 '18 at 20:48
Arthur
111k7105186
add a comment |
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1 Answer
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1 Answer
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Let $y_i=x_i+9$. Then
$$
y_1+y_2+y_3+y_4=66\
0leq y_ileq 30
$$
Now find the number of solutions the way you say you know.
answered Dec 10 '18 at 20:48
Arthur
111k7105186
add a comment |
Let $y_i=x_i+9$. Then
$$
y_1+y_2+y_3+y_4=66\
0leq y_ileq 30
$$
Now find the number of solutions the way you say you know.
answered Dec 10 '18 at 20:48
Arthur
111k7105186
add a comment |
Let $y_i=x_i+9$. Then
$$
y_1+y_2+y_3+y_4=66\
0leq y_ileq 30
$$
Now find the number of solutions the way you say you know.
answered Dec 10 '18 at 20:48
Arthur
111k7105186
Let $y_i=x_i+9$. Then
$$
y_1+y_2+y_3+y_4=66\
0leq y_ileq 30
$$
Now find the number of solutions the way you say you know.
answered Dec 10 '18 at 20:48
Arthur
111k7105186
answered Dec 10 '18 at 20:48
Arthur
111k7105186
answered Dec 10 '18 at 20:48
Arthur
111k7105186
answered Dec 10 '18 at 20:48
Arthur
111k7105186
111k7105186
add a comment |
add a comment |
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It is the coefficient of $x^{30}$ in $$frac{x^{-36}(1-x^{30})^4}{(1-x)^4}$$
– Matt Samuel
Dec 10 '18 at 20:44
@MattSamuel Could you explain this? We have to use sets (finished generating functions chapter).
– Math Newbie
Dec 10 '18 at 20:46
The formula for $N$ you give is the unrestricted value. That is, it would be the answer if the question was how many solutions are there to $x_1+x_2+x_3+x_4=30$ with $x_i≥0$. More work is needed to handle a cap, as in $0≤x_i≤21$.
– lulu
Dec 10 '18 at 20:58
"set theory" has nothing to do with your question.
– Jean Marie
Dec 10 '18 at 21:47
@JeanMarie Ah, sorry, I assumed so because that's our current chapter.
– Math Newbie
Dec 10 '18 at 22:34