:o
so 
and since 
is just 
therefore 
so 
can be defined as a sequence such that 
,
,
.
is such that 
.
for the function 
are perfect squares?
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
be external direct sum of vector spaces 
and 
over a field 
.
and 
and 
is subspaces.
.
be the external direct sum of 
Then 
is
and whose roots are all real.
.
with a finitely generated k-algebra with one generator such that make 
there exist 
such that 
.
then 
then the result from a) may no longer hold.
invertible, for an infinite number of 
:
such that 
is a scalar
![\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}}
\, \mathrm{d}u
\]](http://latex.artofproblemsolving.com/9/1/1/91165e6b531233c56b799ed32ded982d57b6c87a.png)
and 
be positive integers. Prove that![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](http://latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
is not a perfect square.
Y by PikaPika999, Exponent11
Hello!
Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator)
1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.
2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.
3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.
4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.
5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.
6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.
7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.
8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.
9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.
10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
Here are some challenging practice counting problems. Enjoy! (You're allowed to use a calculator)
1.
Yan rolls 9 standard six-sided dice.
What is the probability that at least one pair of dice has a sum of 8?
Round your answer to 3 decimal places.
2.
Each face of a cube is painted one of 5 colors: red, blue, green, yellow, or white.
What is the probability that no two adjacent faces are painted the same color?
Round your answer to 3 decimal places.
3.
You roll 8 standard six-sided dice in a row.
What is the probability that at least one pair of adjacent dice differ by exactly 2?
Round your answer to 3 decimal places.
4.
A 4×4×4 cube (made of 64 mini-cubes) is randomly painted, each mini-cube colored independently either black or white.
What is the probability that at least one mini-cube adjacent to the center mini-cube is black?
Round your answer to 3 decimal places.
5.
Yan rolls 7 dice, each numbered 11 to 88.
What is the probability that at least two dice show the same number?
Round your answer to 3 decimal places.
6.
Each vertex of a cube is randomly colored either red, blue, or green.
What is the probability that there exists at least one face whose four vertices are all the same color?
Round your answer to 3 decimal places.
7.
You roll 6 standard six-sided dice.
What is the probability that the sum of all six dice is divisible by 4?
Round your answer to 3 decimal places.
8.
Each face of a cube is randomly colored red, blue, green, or yellow.
What is the probability that no two opposite faces are painted the same color?
Round your answer to 3 decimal places.
9.
Yan flips a fair coin 12 times.
What is the probability that there is at least one sequence of 4 consecutive heads?
Round your answer to 3 decimal places.
10.
Each edge of a cube is randomly colored either red, blue, or green.
What is the probability that no face of the cube has all three edges the same color?
Round your answer to 3 decimal places.
This post has been edited 3 times. Last edited by mithu542, Apr 29, 2025, 9:30 PM
, and it follows that 
and the final probability is 
.
The distinction between overcount or not is whether arrangements with more sequences then necessary are counted more than once.
by considering one string 
and leaving the rest of the choices. We have 
Similarly, we can arrange multiple 
and 
Now, we have the probability of at least one sequence as 
ways to color the edges, so the denominator of the answer, in lowest terms, should divide 
does not divide 
for variables 
and 
that are not independent.
