MATHEMATICS COLLEGE

A firm has 18 senior and 22 junior partners. A committee of three partners is selected at random to represent the firm at a conference. In how many ways can at least one of the junior partners be chosen to be on the committee?

Answers

Answer 1

Answer:

Answer:

Answer is 24288.

Step-by-step explanation:

Given that there are 18 senior and 22 junior partners.

To find:

Number of ways of selecting at least one junior partner to form a committee of 3 partners.

Solution:

At least junior 1 member means 3 case:

1. Exactly 1 junior member

2. Exactly 2 junior member

3. Exactly 3 junior member

Let us find number of ways for each case and then add them.

Case 1:

Exactly 1 junior member:

Number of ways to select 1 junior member out of 22: 22

Number of ways to select 2 senior members out of 18: 18 $*$ 17

Total number of ways to select exactly 1 junior member in 3 member committee: 22 $*$ 18 $*$ 17 = 6732

Case 2:

Exactly 2 junior member:

Number of ways to select 2 junior members out of 22: 22 $*$ 21

Number of ways to select 1 senior member out of 18: 18

Total number of ways to select exactly 2 junior members in 3 member committee: 22 $*$ 21 $*$ 18 = 8316

Case 3:

Exactly 3 junior member:

Number of ways to select 3 junior members out of 22: 22 $*$ 21 $*$ 20 = 9240

So, Total number of ways = 24288

Answer 2

Answer:

Final answer:

The problem can be solved by finding the total number of ways to form a committee of three from all partners and the ways to form a committee solely from senior partners. Subtracting the number of all-senior committees from the total committees yields the number of committees that include at least one junior partner.

Explanation:

This problem is a combination problem dealing with probability and involves the use of the formula for combinations: C(n, k) = n! / [k!(n-k)!], where n is the total number of options, k is the number of options chosen, and '!' denotes a factorial.

The total number of ways to select 3 partners from the 40 (18 senior + 22 junior) is C(40, 3).

The only scenario where a junior partner is not present in the committee is when all three are senior partners. The number of ways to select 3 senior partners from the 18 available is C(18, 3).

So, to find the number of ways to form a committee with at least one junior partner, subtract the number of all-senior committees from the total number of committees. Therefore, the solution is C(40, 3) - C(18, 3).

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Consider the sequence below. 3, 1, 1/3, 1/9,...

select the explicit function which defines the sequence.

A.) f(n) = 1/3 • 2^(n - 1)

B.) f(n) = 2 • (1/3)^(n - 1)

C.) f(n) = 1/3 • 3^(n - 1)

D.) f(n) = 3 • (1/3)^(n - 1)

Answers

ANSWER

The explicit formula that defines the sequence is
$f(n) = 3 ( (1)/(3) ) ^(n - 1)$

EXPLANATION

The given sequence is
$3,1, (1)/(3) , (2)/(9) ,...$

The first term of this sequence is
$a = 3$
We can find the common ratio by expressing a subsequent term over a previous term and simplifying it.

The common ratio is
$r = (1)/(3)$

The formula for finding the nth term of the given geometric sequence is given by,

$f(n) = a {r}^(n - 1)$

We now substitute the value of the first term and the common ratio in to the above formula to obtain,

$f(n) = 3( (1)/(3) )^(n - 1)$

The correct answer is option D.

Answer:

Step-by-step explanation:

What is the domain of x = 4?

Answers

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined. The range is the set of all valid y values.

Step-by-step explanation:

since x=4 is a vertical line,there is no y-intercerpt and the slope is undefined.

Use the inner product〈f,g〉=∫10f(x)g(x)dxin the vector space C0[0,1] of continuous functions on the domain [0,1] to find 〈f,g〉, ∥f∥, ∥g∥, and the angle αf,g between f(x) and g(x) forf(x)=−10x2−6 and g(x)=−9x−4.〈f,g〉= ,∥f∥= ,∥g∥= ,αf,g .

Answers

Answer:

a) <f,g> = 2605/3

b) ∥f∥ = 960

c) ∥g∥ = 790

d) α = 90

Explanation

a) We calculate <f,g> using the definition of the inner product:

$<f,g> = \int\limits^1_0 {10(-10x^(2) -6)(-9x-4)} \, dx \n \n =\int\limits^1_0 {900x^(3)+400x^(2) +540x+240 } \, dx\n \n = (225x^(4) + (400x^(3) )/(3) + 270x^(2) +240x)\n = (2605)/(3)$

b) How

∥f∥ = <f,f> then:

∥f∥ = $<f,f> = \int\limits^1_0 {10(-10x^(2) -6)(-10x^(2) -6)} \, dx \n \n =\int\limits^1_0 {1000x^(4)+1200x^(2) + 360} \, dx\n \n = (200x^(5) + 400x^(3) + 360x)\n = 960$

∥g∥ = <g,g>

∥g∥ = $<g,g> = \int\limits^1_0 {10(-9x-4)(-9x-4)} \, dx \n \n =\int\limits^1_0 {810x^(2)+720x + 160} \, dx\n \n = (270x^(3) + 360x^(2) + 160x)\n = 790$

d) Angle between f and g

<f,g> = ∥f∥∥g∥cosα

Thus

$\alpha = cos^(-1)((2605/3)/((790)(960)) )\n\n\alpha = 90$

Final answer:

The answer to this problem involves applying integrals, norms, and concepts of angles between vectors to the functions f(x) and g(x). The INNER PRODUCT is the integral of the products of the two functions, the norms are the square roots of the inner products of the functions with themselves, and the angle between the functions is calculated using the dot product and norms.

Explanation:

To find the inner product 〈f,g〉, the norms ∥f∥ and ∥g∥, and the angle αf,g between the functions f(x)=−10x2−6 and g(x)=−9x−4, we'll apply concepts from vector calculus. The inner product (also known as the dot product) is the integral from 0 to 1 of the products of the two functions. The norm of a function is the square root of the inner product of the function with itself. The angle between two vectors in a Vector Space, in this case the space of continuous functions C0[0,1], is given by cos(α) = 〈f,g〉/( ∥f∥∙ ∥g∥). Integrating and solving these equations will give us the desired values.

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The y-intercept of a parabola is 1, and its vertex is at (1,0). What function does the graph represent?OA. Rx) = (x - 1)2
OB. Rx) = (x + 1)2
OC. Rx) = -1(x - 1)
OD. Rx) = -1(x + 1)2
Reset
Next

Answers

Considering it's y-intercept and vertex, the equation of the parabola is given by:

$y = (x - 1)^2$

What is the equation of a parabola given it’s vertex?

The equation of a quadratic function, of vertex (h,k), is given by:

$y = a(x - h)^2 + k$

In which a is the leading coefficient.

In this problem, the vertex is (1,0), hence h = 1, k = 0 and:

$y = a(x - 1)^2$

The y-intercept is of 1, hence, when x = 0, y = 1, so:

$y = a(x - 1)^2$

$1 = a(0 - 1)^2$

$a = 1$

Hence, the equation is:

$y = (x - 1)^2$

More can be learned about the equation of a parabola at brainly.com/question/24737967

Answer:

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

Here (h, k) = (1, 0) , thus

y = a(x - 1)² + 0

To find a substitute the coordinates of the y- intercept (0, 1) into the equation

1 = a(- 1)² = a , thus

a = 1

y = (x - 1)² → A

3) g(x) = 4x
h(x) = -3x^2 + 4
Find (g + h)(x)

Answers

Answer:

The answer is the last line below.

Step-by-step explanation:

g(x) = 4x

h(x) = -3x^2 + 4

(g + h)(x) = 4x - 3x^2 + 4

Stephanie was doing a science experiment to see what brand of cereal with raisinsactually had more raisins. She scooped out one cup of cereal and counted 27 raisins. If
there were 12 cups of cereal in the box, predict about how many raisins should there
be in the box?

Answers

Answer:

counted 27 raisins. If

there were 12

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