Set theory, permutations and combinations

What is the formula for combination with repetition allowed?

(r+n-1)!/r!(n-1)!

What is meant by the term mutually exclusive?

2 events are mutually exclusive when they cannot occur simultaneously.

What is an example of a permutation with repetition allowed?

A DNA sequence of length r.

What is the notation for A being a subset of B?

A being a subset of B is denoted A⊆B.

What is a set?

A collection of elements.

What is a proper subset?

A is a proper subset of B only if every element of A is in B and there exists at least 1 element of B that is not in A.

What is a subset?

A is a subset of B only if every element of A is in B.

What is the notation for the complement of a set A (all of the subsets not including A)?

A'.

Why does combinations with no repetition allowed and choosing all elements become n!?

Because

How can permutations with no repetition be thought of graphically?

Imagine a grid of boxes with length r = 3: ☐₁x☐₂x☐₃ = ans In this case, box one contains n objects and each consecutive box contains one less object than the last (n!). However, because you are only choosing r objects, the difference (n-r)! is divded out.

How can permutations with repetition be thought of graphically?

Imagine a grid of boxes with length r = 3: ☐₁x☐₂x☐₃ = ans In this case, each box can be n objects and the answer is n^r.

Can r > n?

No. For example you cant choose 4 from 3 items.

What is an example that distinguishes permutations from combinations?

The example of 2P4 (nucleotides). In the permutation, AT and TA are distinct, whereas in the combination they are the same.

What is the notation for the intersection of sets A and B (containing all of the elements common to both sets)?

The intersection of sets A and B is denoted A∩B.

What is a combination?

The number of subsets of a set n that contain r elements, where the order of the elements is irrelevant.

What is a permutation?

The number of subsets of a set n that contain r elements, where the order of the elements matters.

What is the notation for the union of sets A and B (containing all of the elements in both A and B)?

The union of sets A and B is denoted A∪B.

In combinations and permutations, what are the symbols for the total number of elements and the number of elements chosen?

n and r, respectively.

What is the formula for combination with no repetition allowed and choosing all elements?

n!

What is the formula for permutation with no repetition allowed?

n!/(n-r)!

What is the formula for combination with no repetition allowed?

n!/r!(n-r)!

What is the shorthand for combination with no repetition allowed?

nCr (r combinations of n things).

What is the shorthand for permutation with no repetition allowed?

nPr (r permutations of n things).

What is the formula for permutation with repetition allowed?

n^r.

What is the notation for an element x belonging to a set A?

x∈A.