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Counting teams with Cooper

A team of 3 student volunteers is formed at random from a class of 40. What is the probability that Cooper is on the team?

Solution

Haley and Hugo from 2 groups of 3

The class has 40 students. Suppose the professor chooses 3 students Wednesday at random, and again 3 on Friday. What is the probability that Haley is chosen today and Hugo on Friday?

Solution

(1) Count total outcomes.

Have possible groups chosen Wednesday.

Have possible groups chosen Friday.

Therefore possible groups in total.

(2) Count desired outcomes.

Groups of 3 with Haley are same as groups of 2 taken from others.

Therefore have groups that contain Haley.

Have groups that contain Hugo.

Therefore total desired outcomes.

(3) Compute probability.

Let label the desired event.

Use formula:

Therefore:

Counting VA license plates

A VA license plate has three letters (with no I, O, or Q) followed by four numerals. A random plate is seen on the road.

(a) What is the probability that the numerals are in increasing order?

(b) What is the probability that at least one number is repeated?

Solution

(a)

(1) Count ways to have 4 numerals in increasing order.

Any four distinct numerals have a single order that’s increasing.

There are ways to choose 4 numerals from 10 options.

(2) Count ways to have 3 letters in order except I, O, Q.

26 total letters, 3 excluded, thus 23 options.

Repetition allowed, thus possibilities.

(3) Count total plates with increasing numerals.

Multiply the options:

(4) Count total plates.

Have options for letters.

Have options for numbers.

Thus possible plates.

(5) Compute probability.

Let label the event that a plate has increasing numerals.

Use the formula:

Therefore:

(b)

(1) Count plates with at least one number repeated.

“At least” is hard! Try complement: “no repeats”.

Let be event that no numbers are repeated. All distinct.

Count possibilities:

Total license plates is still .

Therefore, license plates with at least one number repeated:

(2) Compute probability.

Desired outcomes over total outcomes:

Counting out 4 teams

A board game requires 4 teams of players. How many configurations of teams are there out of a total of 17 players if the number of players per team is 4, 4, 4, 5, respectively.

Solution