Training Probability & Combinatorics Combinations & the Binomial Theorem
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Combinations & the Binomial Theorem

22 min Probability & Combinatorics
Combinations count unordered selections: C(n, r) = n!/(r!(n − r)!), often written as 'n choose r.' The binomial theorem expands (a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ, connecting algebra to combinatorics. Pascal's triangle encodes the binomial coefficients, with each entry being the sum of the two entries above it. Useful identities include C(n, r) = C(n, n − r), the hockey-stick identity, and Vandermonde's identity. Combinations with repetition—choosing r items from n types where repeats are allowed—use the formula C(n + r − 1, r), sometimes called 'stars and bars.'

Combinations & the Binomial Theorem

Combination

An unordered selection of $r$ objects from $n$ distinct objects:

$$\binom{n}{r} = C(n,r) = \frac{n!}{r!(n-r)!}$$

Key Properties

$$\binom{n}{0} = \binom{n}{n} = 1 \qquad \binom{n}{r} = \binom{n}{n-r}$$

$$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r} \quad \text{(Pascal's Rule)}$$

Binomial Theorem

$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

Example 1

A committee of 3 is chosen from 10 people. How many committees?

  1. Set up the problem.
  2. $$\binom{10}{3} = \frac{10!}{3! \cdot 7!} = \frac{720}{6} = 120$$
Example 2

Expand $(x+2)^4$.

  1. $$\binom{4}{0}x^4 + \binom{4}{1}x^3(2) + \binom{4}{2}x^2(4) + \binom{4}{3}x(8) + \binom{4}{4}(16)$$
  2. $$= x^4 + 8x^3 + 24x^2 + 32x + 16$$
Example 3

How many 5-card hands from a 52-card deck?

  1. Set up the problem.
  2. $$\binom{52}{5} = 2{,}598{,}960$$

Practice Problems

1. Compute $\binom{7}{3}$.
2. Compute $\binom{10}{2}$.
3. Compute $\binom{8}{8}$.
4. A team of 4 from 12 players. How many teams?
5. Verify $\binom{6}{2} = \binom{6}{4}$.
6. Use Pascal's Rule: $\binom{5}{2} = \binom{4}{1} + \binom{4}{2}$.
7. Expand $(a+b)^3$ using the Binomial Theorem.
8. Find the coefficient of $x^2$ in $(x+3)^5$.
9. $\binom{20}{1}$?
10. A pizza shop offers 8 toppings. How many 3-topping pizzas?
11. Compute $\binom{9}{4}$.
12. How many diagonals does a convex 8-gon have? ($\binom{8}{2} - 8$)
Show Answer Key

1. $35$

2. $45$

3. $1$

4. $\binom{12}{4} = 495$

5. Both equal $15$ ✓

6. $4 + 6 = 10 = \binom{5}{2}$ ✓

7. $a^3 + 3a^2b + 3ab^2 + b^3$

8. $\binom{5}{2} \cdot 3^3 \cdot 1 = 10 \cdot 27 = 270$... wait: coeff of $x^2$ means $\binom{5}{3}3^3 = 10 \cdot 27 = 270$

9. $20$

10. $\binom{8}{3} = 56$

11. $126$

12. $28 - 8 = 20$

📐 Binomial Coefficient & Expansion
C(n,k)
Pascal's row n
2ⁿ (row sum)
Counting Principles & Permutations Probability Fundamentals