Question

How many solutions are there to equation x1 + x2 + x3 + x4 = 15 where xi , for i = 1, 2, 3, 4, is a nonnegative integer and

(a) x1 > 1?

(b) xi ≥ i, for i = 1, 2, 3, 4?

(c) x1 ≤ 13?

Answer #1

How many different integer solutions are there to the equation
x1 + x2 + x3 + x4 + x5 + x6 + x7 = 23, 0 ≤ xi ≤ 9 ?
(a) (2 points) Solve the problem by using Inclusion-Exclusion
Formula. (b) (2 points) Check whether your solution obtained from
part
(a) is right by using the generating function method.

Find the number of solutions to
x1+x2+x3+x4=16 with
integers x1 ,x2, x3, x4
satisfying
(a) xj ≥ 0, j = 1, 2, 3, 4;
(b) x1 ≥ 2, x2 ≥ 3, x3 ≥ −3,
and x4 ≥ 1;
(c) 0 ≤ xj ≤ 6, j = 1, 2, 3, 4

How many integral solutions of x1 + x2 + x3 + x4 = 33 satisfy 4
≥ x1 ≥ 2, x2 ≥ 0, x3 ≥−5, and x4 ≥ 7?

How many integer solutions are there to
x1+x2+x3+x4= 100 with
all of the following constraints:
10 ≤ x1 , 0≤ x2 < 20 , 0 ≤
x3 < 40 , 10 ≤ x4< 50

2. Find the number of integer solutions to x1 + x2 + x3 + x4 +
x5 = 50, x1 ≥ −3, x2 ≥ 0, x3 ≥ 4, x4 ≥ 2, x5 ≥ 12.

How many non-negative integer solutions are there to
x1+x2+x3+x4+x5 = 60
(a) where x1 <= 17 and x2 <= 17
(b) where x1 <= 17 and x2 <= 17 and x3 <= 17 and x4 <=
17
Side question: is there any reason why, for (a), that we can't just
give x1 and x2 both 18 "stars" (from the sticks and stars
representation of the problem) and then calculate the number of
ways to distribute the remaining 60 - (18...

Find the number of 5-lists of the form (x1, x2, x3, x4, x5),
where each xi is a nonnegative integer and x1 + x2 + x3 + x4 + 3x5
= 12.
Does this have to be answered by cases or is there a systematic
way to determine all of the possibilities?

How many nonnegative integer solutions are there to x1 + x2 + .
. . + x5 = 20 with xi less than or equal to 10?

What is the generating function for the number of non-negative
integer solutions to
x1 + x2 + x3 + x4 + x5 = 50
if:
1.) There are no restrictions
2.) xi >= 2 for all i
3.) x1 <= 10
4.) xi <= 12 for all i
5.) if x1 is even

Give augmented matrix for this system. Find all solutions to
this system. Indicate all parameters.
x1-x2+x3+x4=1
2x2+3x3+4x4=2
x1-x2+2x3+3x4=3
x1=? x2=? x3=? x4=?

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