Permutations and Combinations

• Permutations (Order matters)
  • Replacement:
  • No replacement
    • Distinct objects:
      • Or when
    • Non-distinct Objects
      • when where are the quantity of duplicate events
• Combinations (Order does not matter)

A permutation is an ordered arrangement of objects selected from a set

The number of permutations for objects in "positions" with replacement is:

In how many ways can three different awards be distributed among 20 students in the following situations:

1. No student may receive more than 1 award (no replacement)
2. There is no limit on the number of awards won by any student (with replacement)

An ATM with 4 digit passcode. Calculate the probability that a friend's ATM pin starts with 8 and ends with 2.

solution
Not all 4-digit passcodes are equally likely. However, we can just assume they are.

We have:

Possibilities for each position:

So

The number of permutations for objects for "positions" without replacement is:

for

(See Factorial)

Determine the number of 4 letter permutations of the letters in the word "PROBLEM"

solution

Create a 4-letter "word" using A, B, C, and D. In how many ways can all the letters be arranged without replacement?

So ways.

3 Letter word?

A three-digit number is randomly constructed from , where each digit can only be used once. Find the probability that the number starts with a 1.

solution

or you could just say, since we only care about the first number, and we don't know about the other digits.

Twelve students have signed up to serve on the yearbook committee

1. How many ways can the staff advisor choose three people to act as publisher, photographer, and editor?

2. What is the probability that your mom is elected for role 1 and your dad is elected for role 2?

The number of permutations of objects of which are alike one type, are alike another type, etc, is:

You can arrange BOP in different ways: BOP, BPO, OBP, OPB, PBO, POB
If you try to arrange BOB, you find that only BOB, OBB, and BBO are distinct; the rest would be duplicate
Hence the number of arrangements of BOB is .

How many distinct arrangements of the letters in "STATEMENT" can be made?

solution
total letters, "T"s and "E"s

Random words are constructed using the letters from the word MISSISSIPPI. Each letter can only be used once. Find the probability that all the S's are together.

solution

• S: 4
• I: 4
• P: 2
• M: 1

which is the formula for permutations of non-distinct objects.
now treat the 4 S's as one "super S":

so

The number of circular permutations of objects is:

7 students show up for lunch. How many permutations are possible if:

1. They line up for a photograph

2. They sit at a round table

3. They sit at a round table, but Imposter and Pissface must sit together

A combination does not take into account order, whereas permutations do

The number of combinations of objects, selected at a time is:

(see binomial coefficient)

Since combinations are just permutations without taking into account order, there are more permutations than combinations. That is, .

A committee of 6 people is formed from 5 teachers and 8 students

1. How many possible committees are there?

2. Determine the probability that there are exactly 2 students on the committee

3. Determine the probability that there are more teachers than students on the committee

4. Determine the probability that at least 1 teacher is on the committee

In Lotto 6/49, determine the odds in favour of correctly guessing three of the six numbers

Which is