Combinations
Understand how to count selections where order does not matter, using the nCr formula and its connection to Pascal's triangle.
What Are Combinations?
A combination is a selection of items from a larger set where the order does not matter. For example, choosing 3 students from a class of 10 to form a committee is a combination problem because it does not matter in which order the students are picked.
This is different from a permutation, where order matters (e.g., choosing a president, secretary, and treasurer from the same group of 10 would be a permutation).
The nCr Formula
Number of combinations of r items from n:
nCr = n!/r!(n - r)!
n = total number of items to choose from
r = number of items being chosen
Permutations vs Combinations
Permutations (order matters)
Selecting A then B is different from B then A.
Formula: nPr = n! / (n-r)!
Combinations (order does not matter)
Selecting A then B is the same as B then A.
Formula: nCr = n! / [r!(n-r)!]
Pascal's Triangle Connection
The values of nCr appear as entries in Pascal's triangle. Each row corresponds to a value of n, and each position within the row corresponds to a value of r.
Pascal's Triangle (rows 0 to 5)
Row 4, position 2 gives 4C2 = 6. Row 5, position 2 gives 5C2 = 10.
Key property: nCr = nCn-r. This symmetry is visible in Pascal's triangle, where each row is a palindrome.
Applying the Formula
To decide whether a problem requires combinations, ask: "Does the order of selection matter?" If the answer is no, use combinations.
Common Combination Scenarios
- ●Choosing a committee or team from a group
- ●Selecting toppings for a pizza
- ●Picking lottery numbers (without order)
- ●Choosing cards from a deck (e.g., a poker hand)
Key Vocabulary
Combination
A selection of items where order does not matter. Denoted nCr or C(n, r).
Factorial (n!)
The product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Pascal's Triangle
A triangular array where each entry equals the sum of the two entries directly above it. Row n contains the values nC0 through nCn.
Permutation
An arrangement of items where order matters. Denoted nPr.
Worked Examples
Calculate 6C2.
Step 1: Write out the formula: 6C2 = 6! / (2! x 4!)
Step 2: Expand: = (6 x 5 x 4!) / (2 x 1 x 4!)
Step 3: Cancel 4!: = (6 x 5) / 2 = 30/2
Answer: 6C2 = 15
A team of 3 is to be chosen from 8 athletes. How many different teams are possible?
Step 1: Order does not matter (it's a team), so use combinations. n = 8, r = 3.
Step 2: 8C3 = 8! / (3! x 5!) = (8 x 7 x 6) / (3 x 2 x 1)
Step 3: = 336/6
Answer: There are 56 different teams possible.
Show that 5C2 = 5C3.
Step 1: 5C2 = 5! / (2! x 3!) = (5 x 4) / (2 x 1) = 10
Step 2: 5C3 = 5! / (3! x 2!) = (5 x 4) / (2 x 1) = 10
Answer: Both equal 10, confirming the symmetry property nCr = nCn-r.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the value of 7C3?
Question 2
A pizza shop offers 10 toppings. How many ways can you choose 4 toppings?
Question 3
Which row of Pascal's triangle contains the value 6C2 = 15?
Question 4
If nC2 = 28, what is the value of n?
Question 5
A committee of 5 is to be chosen from 12 people. How many different committees are possible?
Key Concepts Summary
- ● A combination counts selections where order does not matter.
- ● The formula is nCr = n! / [r!(n - r)!].
- ● The symmetry property states nCr = nCn-r.
- ● Combination values appear as entries in Pascal's triangle.
- ● Combinations give fewer arrangements than permutations because order is not counted.