Member: sarafoster

0 Questions
9 Answers

-9

Consider the set of all possible pairs of points that can be chosen from the given set of distinct points on the circle. There are $\binom{n}{2}$ such pairs, where $n$ is the total number of distinct points on the circle. Each pair of points corresponds to a chord that can be drawn between them.

Now, we want to find the probability that when we choose four chords at random, they form a bow-tie quadrilateral. To do this, we can count the total number of ways to choose four chords and the number of ways to choose four chords that form a bow-tie quadrilateral.

The total number of ways to choose four chords is $\binom{\binom{n}{2}}{4}$, which is the number of ways to choose 4 pairs of points from the set of $\binom{n}{2}$ pairs.

To count the number of ways to choose four chords that form a bow-tie quadrilateral, we first choose two chords that intersect. There are $\binom{4}{2} = 6$ ways to do this. Once we have chosen the intersecting chords, we need to choose two more chords that do not intersect either of the first two chords. There are $\binom{\binom{n}{2}-4}{2}$ ways to do this.

Therefore, the number of ways to choose four chords that form a bow-tie quadrilateral is $6 \times \binom{\binom{n}{2}-4}{2}$.

Hence, the probability that four randomly chosen chords form a bow-tie quadrilateral is:

$$ \frac{6 \times \binom{\binom{n}{2}-4}{2}}{\binom{\binom{n}{2}}{4}} $$

Simplifying the expression, we get:

$$ \frac{3(n-4)}{(n-1)(n-2)(n-3)} $$

where $n$ is the total number of distinct points on the circle.

sarafosterFeb 21, 2023

-9

-1

(a) Let's first find out how much magical syrup is in 300 mL of blue potion. Since the blue potion is 15% magical syrup by volume, the amount of magical syrup in 300 mL of blue potion is 0.15 x 300 = 45 mL.

Now, let's assume that we add x mL of red potion to this mixture. The resulting potion will have a total volume of 300 + x mL, and its magical syrup content will be (45 mL + 0.5x mL) / (300 mL + x mL). We want this to be equal to 20% (or 0.2), so we can set up the following equation:

(45 mL + 0.5x mL) / (300 mL + x mL) = 0.2

Solving for x, we get:

x = 150 mL

Therefore, 150 mL of red potion must be added to 300 mL of blue potion to produce a potion that is 20% magical syrup by volume.

(b) Let's assume that we mix x mL of red potion with y mL of blue potion to produce 180 mL of a potion that is 40% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.4(180 mL)

Total volume: x mL + y mL = 180 mL

Simplifying the first equation, we get:

0.5x + 0.15y = 72

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = 180 - x

Substituting this into the first equation, we get:

0.5x + 0.15(180 - x) = 72

Solving for x, we get:

x = 120 mL

Substituting this into the equation y = 180 - x, we get:

y = 60 mL

Therefore, 120 mL of red potion and 60 mL of blue potion can be combined to produce 180 mL of a potion that is 40% magical syrup by volume.

(c) Let's assume that we mix x mL of red potion with y mL of blue potion to produce a potion that is 25% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.25(x+y) mL

Total volume: x mL + y mL = some value

We have two equations and two unknowns, so we can solve for x and y. Simplifying the first equation, we get:

0.25x - 0.1y = 0

Substituting the second equation into this equation, we get:

0.25x - 0.1(x + y) = 0

Simplifying, we get:

0.15x - 0.1y = 0

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = some value - x

Substituting this into the first equation, we get:

0.15x - 0.1(some value - x) = 0

Simplifying, we get:

0.25x - 0.1(some value) = 0

Therefore, we can see that there is no solution that satisfies these equations for any positive values of x and y. So there is no combination of red

sarafosterFeb 20, 2023

-9

The terms in the expansion of $(1+ax)^n$ can be found using the binomial theorem, which states that

$$(1+ax)^n = \sum_{k=0}^n \binom{n}{k} (ax)^k$$

where $\binom{n}{k}$ is the binomial coefficient.

To find the coefficient of the $x^3$ term, we need to find the value of $k$ that satisfies $k \geq 3$ and $n-k=3$. Solving $n-k=3$ for $k$, we get $k=n-3$. Thus, we need to find the coefficient of $(ax)^{n-3}$ in the expansion. Using the formula for the binomial coefficient, we have

$$\binom{n}{n-3} = \frac{n!}{(n-3)!(3)!} = \frac{n(n-1)(n-2)}{6}$$

Multiplying this by $a^{n-3}$ and simplifying, we get the coefficient of $x^3$:

$$c = \frac{n(n-1)(n-2)}{6} a^{n-3}$$

Therefore, in the given expression, $c=\boxed{840}$, since the first three terms of the expansion have been provided.

sarafosterFeb 15, 2023

Username	sarafoster
Score	-9
Membership
Stats
Questions	0
Answers	9