How Many Ways Can You Arrange a Pharmaceutical Board and Solve Math Problems?

[sjaguar13]

1) The board of directors of a pharmaceutical corporation has 10 members. An upcoming stockholder's meeting is scheduled to approve a new slate of company officers (chosen from the 10 board members).

A) 4 (Presendent, Vice Presendent, secretary, and treasurer) positions needs filled. How many possible ways are there to do it?

10 x 9 x 8 x 7

B) Three members of the board of directors are physicians. How many slates from part (A) have a physician nominated for presendency?

3 x 9 x 8 x 7 ?

C) How many slates have exactly one physician?

3 x 7 x 6 x 5

D) How many slates have at least one physician?

3 x 9 x 8 x 7


2) There are 3 cities: a, b, c. City a has two roads that go to C and 4 roads that go to b. City b has 3 roads that go to c.

A) How many ways can you get from a to c?

2 + (4x3)

B) How many round trips from a to c are there such that the return trip is at least partially different than the route taken to get there?

(2 + (4x3)) x (2 + (4x3)) - (2 + (4x3)) ?


3) If n is a positive integer and n is greater than 1, prove that c(n, 2) + c(n-1/2) is a perfect square.

I have no idea


4) A gym coach must select 11 seniors to play on a football team. If he can make his selection in 12,376 ways, how many seniors are eligible to play?

n! / (11! x (n-11)!) = 12,376
n! / (n-11)! = 12,376(11!)
That's all I got


5) How many ways can 10 identical dimes be distributed among five children if:

A) There are no restrictions?

c(14,10)

B) Each child gets at least one dime?

c(9,5)

C) The oldest child gets at least two dimes?

c(12,8)

I know those are the answers, but I don't know why.


6) Determine the number of integer solutions of:
x1 + x2 + x3 + x4 = 32
where:

A) xi >= 0, 1<= i <=4

c(35, 32)

B) xi > 0, 1<= i <=4

c(31, 28) Why?

C) x1, x2 >= 5, x3, x4 >= 7

c(11, 8) Why?

 

[lightgrav]

In 1c, it looks like you're counting
number of Dr.President slates (from B)
that have no other Dr. on it. I would count
yours + (n Dr. n n) + (n n Dr. n) + (n n n Dr.)

In 5, line up the kids and the dimes in a ring, so
each kid gets the dimes clockwise of them.
Now, how many ways can you arrange 15 things
(10 dimes + 3 kids) into the 15 spots on the ring?
If each kid already has been given a dime,
there's 5 dimes left (and 10 spots on the ring).

 

[quicknote]

I would first like to commend the student for showing their work and attempting to solve these problems. It is important to always show your work and reasoning when solving math problems.

Now, let's address each question individually:

1) A) The number of ways to fill the 4 positions is 10 x 9 x 8 x 7, because for the first position, there are 10 choices, then 9 for the second, 8 for the third, and 7 for the fourth.

B) The number of slates from part (A) with a physician nominated for presidency is 3 x 9 x 8 x 7, because for the first position, there are 3 choices (since there are 3 physicians), then 9 for the second, 8 for the third, and 7 for the fourth.

C) The number of slates with exactly one physician is 3 x 7 x 6 x 5, because for the first position, there are 3 choices, then 7 for the second (since there are 7 non-physicians left), 6 for the third, and 5 for the fourth.

D) The number of slates with at least one physician is 3 x 9 x 8 x 7, because we can use the same reasoning as in part (B), but this time we don't have to exclude the possibility of no physicians being nominated.

2) A) The number of ways to get from city a to c is 2 + (4x3), because there are 2 roads from a to c and 4 roads from a to b, and for each of those 4 roads, there are 3 roads from b to c.

B) The number of round trips from a to c such that the return trip is at least partially different is (2 + (4x3)) x (2 + (4x3)) - (2 + (4x3)), because we can use the same reasoning as in part (A), but this time we need to exclude the possibility of taking the same route back.

3) To prove that c(n, 2) + c(n-1/2) is a perfect square, we need to show that it can be written as the square of an integer. We can rewrite c(n, 2) as n! / (2! x (n-2)!