What is a Binomial Coefficient?
The binomial coefficient, denoted as C(n,k) or "n choose k", represents the number of ways to choose k items from a set of n items without regard to order. It's a fundamental concept in combinatorics and probability theory.
Common Applications
- Probability Calculations: Determining the likelihood of specific outcomes in binomial experiments
- Combinatorics: Counting combinations in various mathematical problems
- Pascal's Triangle: Each entry is a binomial coefficient
- Binomial Expansion: Coefficients in the expansion of (a + b)ⁿ
Formula
The binomial coefficient is calculated using the formula:
$$
C(n,k) = \frac{n!}{k! \times (n-k)!}
$$
Where:
- n = Total number of items
- k = Number of items to choose
- n! = n factorial (n × (n-1) × (n-2) × ... × 1)
Calculation Example
Given:
- n = 7 (total items)
- k = 4 (items to choose)
Calculation:
$$
C(7,4) = \frac{7!}{4! \times (7-4)!} = \frac{7!}{4! \times 3!}
$$
$$
= \frac{5040}{24 \times 6} = \frac{5040}{144} = 35
$$
Result: There are 35 different ways to choose 4 items from a set of 7 items.
Properties of Binomial Coefficients
Symmetry Property
$$C(n,k) = C(n, n-k)$$
Choosing k items is the same as choosing which (n-k) items to leave out.
Sum Property
$$C(n,0) + C(n,1) + ... + C(n,n) = 2^n$$
The sum of all binomial coefficients for a given n equals 2ⁿ.
Pascal's Identity
$$C(n,k) = C(n-1,k-1) + C(n-1,k)$$
Each binomial coefficient is the sum of the two directly above it in Pascal's Triangle.
Practical Examples
Example 1: Lottery Combinations
How many ways can you choose 6 numbers from 49?
$$
C(49,6) = \frac{49!}{6! \times 43!} = 13,983,816
$$
Example 2: Team Selection
How many ways can you select 5 players from a roster of 12?
$$
C(12,5) = \frac{12!}{5! \times 7!} = 792
$$
Example 3: Probability in Coin Flips
What's the number of ways to get exactly 3 heads in 5 coin flips?
$$
C(5,3) = \frac{5!}{3! \times 2!} = 10
$$
Important Considerations
- Order doesn't matter: C(n,k) counts combinations, not permutations
- Valid range: k must be between 0 and n (inclusive)
- Edge cases: C(n,0) = 1 and C(n,n) = 1 for all n ≥ 0
- Large numbers: Factorials grow very quickly; calculations may be limited for large n
- Alternative notation: Also written as (n k) or nCk