Complete Guide to nCr Formulas and Combinations

Introduction to nCr (Combinations)

The nCr formula, also known as the combination formula, calculates the number of ways to choose r items from n items where order does not matter. This fundamental concept is essential in combinatorics, probability theory, and statistical analysis.

Core Formulas

Formula Type	Mathematical Expression	Alternative Notation	Explanation
Basic nCr Formula	nCr = n! / (r! × (n-r)!)	C(n,r) = n! / (r! × (n-r)!)	Number of combinations of n items taken r at a time
nPr Formula	nPr = n! / (n-r)!	P(n,r) = n! / (n-r)!	Number of permutations of n items taken r at a time
Relationship Formula	nCr = nPr / r!	C(n,r) = P(n,r) / r!	Combinations equals permutations divided by r factorial
Factorial Definition	n! = n × (n-1) × (n-2) × … × 2 × 1	0! = 1 (by definition)	Product of all positive integers up to n

Essential Properties and Special Cases

Property/Case	Formula	Explanation	Example
Symmetry Property	nCr = nC(n-r)	Choosing r items equals choosing (n-r) items to exclude	5C2 = 5C3 = 10
Boundary Cases	nC0 = nCn = 1	Only one way to choose nothing or everything	7C0 = 7C7 = 1
Single Selection	nC1 = n	n ways to choose one item from n items	8C1 = 8
Pascal’s Identity	nCr = (n-1)Cr + (n-1)C(r-1)	Used to construct Pascal’s triangle	5C2 = 4C2 + 4C1
Sum Property	Σ(nCr) = 2ⁿ (r=0 to n)	Sum of all combinations equals 2 to the power n	3C0 + 3C1 + 3C2 + 3C3 = 8 = 2³

Probability Applications

Application	Formula	Use Case	Example
Basic Probability	P(Event) = nCr / Total Combinations	Calculating probability using combinations	P(2 heads in 4 flips) = 4C2 / 2⁴
Binomial Probability	P(X = k) = nCk × p^k × (1-p)^(n-k)	Probability of exactly k successes in n trials	P(exactly 3 successes in 10 trials)
Hypergeometric	P(X = k) = (KCk × (N-K)C(n-k)) / NCn	Sampling without replacement	Drawing cards from a deck

Advanced Relationships

Relationship	Formula	Mathematical Context
Binomial Theorem	(a + b)ⁿ = Σ(nCr × aⁿ⁻ʳ × bʳ)	Expansion of binomial expressions
Vandermonde’s Identity	(m+n)Cr = Σ(mCk × nC(r-k))	Sum over all valid k values
Chu-Vandermonde	Σ(mCr × nCs) = (m+n)C(r+s)	When summing over specific ranges

Computational Formulas for Large Numbers

Method	Formula	Advantage
Multiplicative Form	nCr = (n × (n-1) × … × (n-r+1)) / (r × (r-1) × … × 1)	Avoids large factorial calculations
Recursive Formula	nCr = (n × (n-1)C(r-1)) / r	Efficient for sequential calculations
Logarithmic Form	log(nCr) = log(n!) – log(r!) – log((n-r)!)	For very large numbers to prevent overflow

Common Mistakes

Error Type	Incorrect	Correct	Note
Order Confusion	Using nPr for unordered selection	Use nCr for combinations	Order doesn’t matter in combinations
Zero Factorial	0! = 0	0! = 1	By mathematical convention
Negative Values	nCr where r > n	nCr = 0 when r > n	Cannot choose more than available
Non-integer Values	Using decimals for n or r	Only use non-negative integers	Combinations require whole numbers

Quick Reference Examples

Basic Calculations

• 5C3 = 5!/(3! × 2!) = (5 × 4)/(2 × 1) = 10
• 7C2 = 7!/(2! × 5!) = (7 × 6)/(2 × 1) = 21
• 10C0 = 1 (choosing nothing)
• 10C10 = 1 (choosing everything)

Real-World Applications

• Team Selection: From 15 players, choose 11 for a team = 15C11 = 1,365
• Card Combinations: Choose 5 cards from 52 = 52C5 = 2,598,960
• Committee Formation: Choose 4 members from 20 people = 20C4 = 4,845

Memory Aids and Tips

1. “Choose, don’t arrange” – Remember nCr is for selection without regard to order
2. Symmetry shortcut – If r > n/2, calculate nC(n-r) instead for easier computation
3. Pascal’s Triangle – Each entry is the sum of the two entries above it
4. Multiplication principle – For sequential choices, multiply combinations

Differences: nCr vs nPr

Aspect	nCr (Combinations)	nPr (Permutations)
Order	Order does NOT matter	Order DOES matter
Formula	n!/(r!(n-r)!)	n!/(n-r)!
Result	Always ≤ nPr	Always ≥ nCr
Example	Selecting team members	Arranging people in line

Frequently Asked Questions (FAQs)

Q. What is the difference between nCr and nPr?

The fundamental difference lies in whether order matters:

• nCr (Combinations): Order does NOT matter. Used when selecting items where arrangement is irrelevant.
  • Example: Choosing 3 students from 10 for a committee = 10C3 = 120
• nPr (Permutations): Order DOES matter. Used when arrangement is important.
  • Example: Arranging 3 students in ranked positions = 10P3 = 720

Relationship: nPr = nCr × r! (permutations are always greater than or equal to combinations)

Q. How do I calculate nCr without a calculator?

Use the multiplicative method for efficient manual calculation:

Step-by-step approach:

1. Write the multiplication form: nCr = (n × (n-1) × … × (n-r+1)) / (r × (r-1) × … × 1)
2. Cancel common factors before multiplying

Example: Calculate 8C3

• Formula: (8 × 7 × 6) / (3 × 2 × 1)
• Cancel: (8 × 7 × 6) / 6 = 8 × 7 / 1 = 56

Q. Why is 0! (zero factorial) equal to 1?

0! = 1 by mathematical convention and logical consistency:

Logical reasons:

• Empty product definition: The product of no numbers equals 1 (multiplicative identity)
• Consistency with nCn: For nCn = n!/(n! × 0!) to equal 1, we need 0! = 1
• Recursive property: Since n! = n × (n-1)!, then 1! = 1 × 0!, so 0! must equal 1

Practical application: nC0 = n!/(0! × n!) = 1 (there’s exactly one way to choose nothing)

Q. When should I use nCr in real-life situations?

Use nCr whenever you need to select or choose items where order doesn’t matter:

Common applications:

• Lottery systems: Choosing 6 numbers from 49 = 49C6
• Team formation: Selecting 11 players from 15 for a football team = 15C11
• Menu combinations: Choosing 3 toppings from 10 available = 10C3
• Card games: Selecting 5 cards from a 52-card deck = 52C5
• Committee selection: Choosing 4 members from 20 people = 20C4
• Investment portfolios: Selecting 5 stocks from 50 options = 50C5

Key indicator: If rearranging your selection doesn’t create a new outcome, use nCr.

Q. What happens when r > n (r is greater than n)?

When r > n, nCr = 0 (mathematically undefined in standard contexts)

Logical explanation:

• You cannot choose more items than what’s available
• Example: You can’t select 5 apples from a basket of 3 apples

Mathematical handling:

• Standard definition: nCr = 0 when r > n
• The formula n!/(r!(n-r)!) becomes invalid because (n-r)! would be the factorial of a negative number

Important note: In standard combinatorics, both n and r must be non-negative integers with r ≤ n.

Q. How do I solve equations like nCr = nPr?

Answer: Use the relationship nCr = nPr / r! and solve algebraically:

Given: nCr = nPr

Solution process:

1. Substitute formulas: n!/(r!(n-r)!) = n!/(n-r)!
2. Simplify: 1/r! = 1
3. Therefore: r! = 1
4. Answer: r = 0 or r = 1

Verification:

• When r = 0: nC0 = nP0 = 1
• When r = 1: nC1 = nP1 = n

Insight: nCr equals nPr only when selecting 0 or 1 item, because arrangement doesn’t create new outcomes.

Q. Can nCr give a decimal or fractional answer?

No, nCr always gives a whole number (non-negative integer) when n and r are valid inputs.

Mathematical guarantee:

• Combinations count distinct selections, which must be whole numbers
• The formula n!/(r!(n-r)!) is designed to always yield an integer when 0 ≤ r ≤ n

Common misconception: Students sometimes get decimals due to:

• Calculation errors (incorrect order of operations)
• Using non-integer values for n or r (invalid inputs)
• Incorrect formula application

Validation check: If you get a decimal answer, recheck your calculation – you’ve made an error.

Q. What is Pascal’s Triangle and how does it relate to nCr?

Pascal’s Triangle is a triangular array where each number equals the sum of the two numbers above it, and each entry represents a combination value.

Structure and nCr relationship:

Row 0: 1 (0C0)
Row 1: 1 1 (1C0, 1C1)
Row 2: 1 2 1 (2C0, 2C1, 2C2)
Row 3: 1 3 3 1 (3C0, 3C1, 3C2, 3C3)
Row 4: 1 4 6 4 1 (4C0, 4C1, 4C2, 4C3, 4C4)

Main properties:

• Row n, Position r = nCr
• Pascal’s Identity: nCr = (n-1)Cr + (n-1)C(r-1)
• Binomial expansion: Row n gives coefficients of (a+b)ⁿ
• Quick calculation: Use the triangle to find combinations without direct calculation