Question

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.

Answer #1

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

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 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?

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...

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

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

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

Find the number of integer solutions to x1+x2+x3=20 given the
following restrictions:
(A) x1>=3, x2>=2,x3>=5
(B) x1>=0, x2>=0, x3<=6

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 9 minutes ago

asked 13 minutes ago

asked 13 minutes ago

asked 26 minutes ago

asked 38 minutes ago

asked 49 minutes ago

asked 53 minutes ago

asked 53 minutes ago

asked 56 minutes ago

asked 56 minutes ago

asked 58 minutes ago