Permutations and Combinations

A chamber of commerce board has seven total members, drawn from a pool of twenty candidates.   There are two stages in the board's election process.  First, a president, secretary, and treasurer are chosen.  After that, four members are chosen to be “at large” without any specific title or district.  How many possible boards could be chosen?

• 390,700,800

• 16,279,200

• 10,465,200

• 5,426,400

In how many ways can 6 differently colored rose bushes be planted in a row that only has room for 4 bushes?

• 240

• 220

• 360

• 500

What is the minimum amount of handshakes that can occur among fifteen people in a meeting, if each person only shakes each other person's hand once?

• 105

• 220

• 250

• 330

There are 20 people eligible for town council, which has three elected members.

Quantity A
The number of possible combinations of council members, presuming no differentiation among office-holders.

Quantity B

The number of possible combinations of council members, given that the council has a president, vice president, and treasurer.

• Quantity A is greater.

• Quantity B is greater.

• The quantities are equal.

• The relationship cannot be determined from the information given.

How many different license passwords can one make if said password must contain exactly 6 characters, two of which are distinct numbers, another of which must be an upper-case letter, and the remaining 3 can be any digit or letter (upper- or lower-case) such that there are no repetitions of any characters in the password?

• 365580800

• 231

• None

• 456426360

Ana is dressing warm for the winter. She'll be layering three shirts over each other, and two pairs of socks. If she has fifteen shirts to choose from, along with ten different kinds of socks, how many ways can she layer up?

• 245700
  

• 2820
  

• 20475
  

• 6360

There are x children and y chairs in a room where x and y are prime numbers. In how many ways can the x children be seated in the y chairs (assuming that each chair can seat exactly one child)?

Statement 1: x + y = 12
Statement 2: There are more chairs than children.

• statement 1 alone is sufficient!
  

• statement 2 alone is sufficient!
  

• Both statements are needed
  

• Data inadequate

A foot race will be held on Saturday. How many different arrangement of medal winners are possible?

1. Statement 1: Medals will be given for 1st, 2nd and 3rd place.

2. Statement 2: There are 10 runners in the race.

• statement 1 alone is sufficient!
  

• statement 2 alone is sufficient!
  

• Both statements are needed
  

• Data inadequate

If the letters of the word CHASM are rearranged to form 5 letter words such that none of the word repeat and the results arranged in ascending order as in a dictionary what is the rank of the word CHASM?

The President has to choose from six members of congress to serve on a committee of three possible members. How many different groups of three could he choose?