1. (12 points) The tennis team at Taft High School has 8 players, and the tennis team at McKinley High School has 7 players. Rebecca is on the team at Taft, and her sister Leah is on the team at McKinley. The two teams are going to play a tournament with 4 rounds. In each round, one player from Taft will play a match against one player from McKinley. Each player can play in at most one match. A schedule for the tournament consists of an ordered list of the names of the players from each school who will play in each of the four rounds. The schedule does not include the results of each match. (a) How many schedules are there? (b) How many schedules include neither sister? (c) How many schedules include both sisters and have them playing against each other? (d) How many schedules include both sisters, but have them playing in different matches?

Answer:

(a) 1,411,200.

(b) 302,400.

(c) 100,800.

(d) 1,008,000.

Step-by-step explanation:

Permutation is the number of ways to select k items from n distinct items in a specific order.

The formula to compute the permutation or arrangement of k items is:

[tex]^{n}P_{k}=\frac{n!}{(n-k)!}[/tex]

(a)

Compute the total number of schedules as follows:

[tex]\text{Total Number of Schedules}=\ ^{8}P_{4}\times \ ^{7}P_{4}[/tex]

                                          [tex]=\frac{8!}{(8-4)!}\times \frac{7!}{(7-4)!}\\\\=\frac{8\times 7\times 6\times 5\times 4!}{4!}\times \frac{ 7\times 6\times 5\times 4\times 3!}{3!}\\\\=(8\times 7\times 6\times 5)\times (7\times 6\times 5\times 4)\\\\=1411200[/tex]

Thus, the total number of schedules are, 1,411,200.

(b)

Compute the number of schedules that includes neither sister as follows:

If neither of sister are included in the schedule then the number of players from the team are 7 and 6 respectively.

[tex]\text{Number of Schedules without the sisters}=\ ^{7}P_{4}\times \ ^{6}P_{4}[/tex]

                                                               [tex]=\frac{7!}{(7-4)!}\times \frac{6!}{(6-4)!}\\\\=\frac{ 7\times 6\times 5\times 4\times 3!}{3!}\times \frac{ 6\times 5\times 4\times 3\times 2!}{2!}\\\\=(7\times 6\times 5\times 4)\times (6\times 5\times 4\times 3)\\\\=302400[/tex]

Thus, the number of schedules that includes neither sister is 302,400.

(c)

Compute the number of schedules including both sisters and have them playing against each other as follows:

If both the sisters are playing against each other then there will be three more places to be filled in both the teams.

And the two sisters can play in any of the 4 matches.

[tex]\text{number of schedules including} \\\text{both sisters against each other}=(\ ^{7}P_{3}\times \ ^{6}P_{3})\times 4[/tex]

                                                [tex]=[\frac{7!}{(7-3)!}\times \frac{6!}{(6-3)!}]\times 4\\\\=[\frac{ 7\times 6\times 5\times 4!}{4!}\times \frac{ 6\times 5\times 4\times 3!}{3!}]\times 4\\\\=[(7\times 6\times 5)\times (6\times 5\times 4)]\times 4\\\\=100800[/tex]

Thus, the number of schedules including both sisters and have them playing against each other is 100,800.

(d)

Compute the number of schedules including both sisters as follows:

[tex]\# \text {Schedules with the Sisters}=\text{Total}\ \#\ \text{Schedules }-\ \# \text {Schedules without the Sisters}[/tex]                                             [tex]=1411200-302400\\=1108800[/tex]

Now compute the number of schedules having the sisters playing in different matches as follows:

[tex]\text{Playing in Different Matches}=\# \text {Schedules with the Sisters}-\text{Playing together}[/tex]

                                              [tex]=1108800-100800\\=1008000[/tex]

Thus, the number of schedules including both sisters, but have them playing in different matches is 1,008,000.

The cost of a computer declined by
$5.75 per day for 9 days. What was the
total change in price after the 9 days?

What are the roots of the polynomial equation x cubed minus 7 x = 6 x minus 12? Use a graphing calculator and a system of equations. –6, 6 –4, 1, 3 –3, –1, 4 1, 3

choose the correct decimal for the expression. seven and ten-thousandths.......7.10, 0.710, 7.0010, 7.010......we chose 7.0010 and it was wrong. help​

What is the answer to 2+2-3+5=

Divide (3x^5-7x^3+5x^2+x-5)/(x^2-1)

Which statement is true?
O A. The absolute value of 3 is -3.
OB. The opposite of 7 is 7.
O C. The absolute value of 5 is equal to the opposite of 5.
O D. The opposite of -4 is
equal to the absolute value of -4.

SOLVE AND JUSTIFY RIGHT ANSWER GETS BRAIN

What is the remainder when you divide
(2x^3– 5x – 7) by (x + 2) ?

a pair of grant high school hockey fans counted the number of games won by the school each year

what was the rate of change between 2014 and 2015​

Maria took a taxicab from her home to the theater
downtown. The taxicab company charges a flat fee of $5.00
plus $0.25 per mile. Which equation represents C, the total
cost of her ride, in terms of m, the length of the trip in
miles?

Please help me on this question ASAP if you do ill give you brainliest!

Mr. Meyer bought a block of fudge that weighed 1/2 of a pound. He cut the fudge into 5 equal pieces. What was the weight of each piece of fudge?

Simplify.
4(1 + 8 . 9) - 62

whats 5.4 times 3.72

I WILL MARK BRAINLEST AND GIVE 100 POINTS! PLZ HELP ME IM STOOPID

957x8=
how much does it equal too?

Could someone help me solve this? Lines AB and CD are parallel. Enter the measures of the three angles in triangle ABF.

How would I solve number six?

On the blueprint of a house,42 millimeters represents 8 meters. The actual length of the living room is 5 meters. What is its length on the blueprint?

A line passes to the points(
(8,-1) and (-4,2)
What is the y-intercept of this line?
-4
-1
1
4

mean median mode 20, 22, 23, 25, 26, 26, 32

I’ll Give BRAINLIEST Just Help Now

Y’all please help me

A contractor cut 28 feet of wire into sections that were feet long. How many pieces of wire did the contractor have?

Helpppppp pleasssseeeeee

If you score higher than 90% on the test, you'll get an A. If you get an A, you can celebrate.
You scored a 92% on the test. What can you conclude using the Law of Syllogism?

A line passes through the point (0, -1) and has a positive slope. Which of these points could that line pass through?
Check all that apply.
(12,3)
(-2,-5)
(-3,1)
(1.15)
(5,-2)

Which table represents a linear function?

Show all work to multiply quantity 6 plus the square root of negative 64 end quantity times quantity 3 minus the square root of negative 16 end quantity

A biologist observed a population of bacteria that grew at a rate expressed by the exponential equation
f(t) = 256e0.06111)
where t is in minutes. How long will it take the population to reach 5 times its initial value?
Round to two decimal places and do not include "t =" in your answer.