Question

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

Answer #1

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 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 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?

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...

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.

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=?

Let X1 , X2 , X3 ,
X4 be a random sample of size 4 from a geometric
distribution with p = 1/3.
A) Find the mgf of Y = X1 + X2 +
X3 + X4.
B) How is Y distributed?

Suppose x1, x2, x3, x4 is linearly independent in V . Prove that
x1 − x2, x2 − x3, x3 − x4, x4 is also linearly independent in V

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

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