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alina1380 [7]
3 years ago
7

The newly elected president needs to decide the remaining 8 spots available in the cabinet he/she is appointing. If there are 14

eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?
Mathematics
1 answer:
Elanso [62]3 years ago
3 0

Answer:

The members of the cabinet can be appointed in 121,080,960 different ways.

Step-by-step explanation:

The rank is important(matters), which means that the order in which the candidates are chosen is important. That is, if we exchange the position of two candidates, it is a new outcome. So we use the permutations formula to solve this quesiton.

Permutations formula:

The number of possible permutations of x elements from a set of n elements is given by the following formula:

P_{(n,x)} = \frac{n!}{(n-x)!}

If there are 14 eligible candidates for these positions (where rank matters), how many different ways can the members of the cabinet be appointed?

Permutations of 8 from a set of 14. So

P_{(14,8)} = \frac{14!}{(14-8)!} = 121,080,960

The members of the cabinet can be appointed in 121,080,960 different ways.

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Triangle ABC has vertices A(-5, -2), B(7, -5), and C(3, 1). Find the coordinates of the intersection of the three altitudes
fiasKO [112]

Answer:

The coordinates of the intersection of the three altitudes = (-3.5, -1)

Step-by-step explanation:

The altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the opposite side.

There are therefore three altitudes possible in a triangle, one from each vertex. All three altitudes always intersect at the same point called the orthocenter of the triangle.

Let the triangle ABC have altitudes AD, BE and CF as shown in the attached image to this solution. Let the orthocentre be O.

The point O is the point where all the coordinates AD, BE and CF meet.

Hence, to obtain the coordinates of O, we just need to equate the equations of two of the lines that serve as the altitude.

Before that, we need to c9mpute the equations of the two altitudes that we will use.

Noting that the altitudes are perpendicular to the sides of the triangle, we can compute the slopes of the altitudes from caldilating the slopes of the sides.

Slope of AB

= (y₂-y₁)/(x₂−x₁)

= (-5 - (-2))/(7 - (-5))

= (-3/12)

= (-1/4)

Slope of its altitude, CF

= -1 ÷ (Slope of AB)

= -1 × (-1/4)

= 4

The equation of CF is given using point C as,

y – y₁ = m(x – x₁)

y - 1 = 4 (x – 3)

y - 1 = 4x - 12

y = 4x + 13

Slope of BC

= (y₂-y₁)/(x₂−x₁)

= (1 - (-5))/(3 - 7)

= (6/-4)

= (-3/2)

Slope of AD

= −1 ÷ (Slope of BC)

= -1 ÷ (-3/2)

= (2/3)

The equation of AD using point A given as,

y – y₁ = m(x – x₁)

y – (-2)) = (2/3) (x – (-5))

y + 2 = (2x/3) + (10/3)

y = (2x/3) + (4/3)

Now equation the equations of the altitudes CF and AD

y = 4x + 13

y = (2x/3) + (4/3)

4x + 13 = (2x/3) + (4/3)

4x - (2x/3) = (4/3) - 13

(10x/3) = (-35/3)

10x = -35

x = -3.5

y = 4x + 13

y = (4×-3.5) + 13 = -14 + 13 = -1

coordinates of the orthocentre of the triangle = (-3.5, -1)

Hope this Helps!!!

6 0
3 years ago
The sum of the ages of the father and son is 44 years .After 8 years the age of the father will be twice the age of the son . Fi
Vadim26 [7]

Step-by-step explanation:

Let the present age of father = x years

and the present age of son = y years

according to the first condition

x + y =44 ........(i)

according to the second condition

x+8 = 2(y+8)

or x+ 8 = 2y + 16

or x- 2y = 8 ............(ii)

solve the equation (i) & (ii) we get

x + y =44

x- 2y = 8

••••••••••••••••••••••••

3y = 36

or y = 12

put the value of y in equation (i), we get

x + 12 =44

or x = 32

so the present age of father = x =32 years

and the present age of son = y = 12 years.

8 0
3 years ago
F x g / _______ = f<br><br> PLEASE HELP I WILL GIVE BRAINLIEST
katrin2010 [14]
Can you show a picture of the problem?
7 0
2 years ago
Math please help tutor :)
Fantom [35]

Answer:

8.5

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Explain and I’ll mark as brainliest too also worth 20 points
USPshnik [31]

Answer:

x = 3

Step-by-step explanation:

The equation is,

16 = 5x+1

Subtract 1 from both sides,

16-1 = 5x

15 = 5x

Divide by 5,

x = 3

7 0
2 years ago
Read 2 more answers
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