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Counting Principles

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Counting Principles

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Counting Principles

🎯 Learning Objectives

Apply the product rule and sum rule for counting
Compute permutations ( $P(n,k)$ ) and combinations ( $C(n,k)$ )
Understand binomial coefficients and Pascal's identity
Distinguish permutations with vs. without repetition
Solve counting problems with multiple cases

1.1 Intuition: Counting Without Listing

Counting is the oldest mathematical art. The key insight: we can count things without listing them by understanding the structure of choices.

🔑 Key Insight: Most counting problems reduce to: "How many ways to make a sequence of choices?"

1.2 Basic Rules

Product Rule

If task

A

can be done in

m

ways and task

B

n

ways, then

A

AND

B

can be done in

m \times n

ways. Example: How many 2-letter words?

26 \times 26 = 676

Sum Rule

If task

A

can be done in

m

ways and task

B

n

ways, and they are disjoint, then

A

B

can be done in

m + n

ways. Example: Choosing a vowel (

5

) or a consonant (

21

) from alphabet:

5 + 21 = 26

1.3 Permutations

Permutation: An ordered arrangement of distinct objects.

P(n, k) = n \times (n-1) \times \cdots \times (n-k+1) = \frac{n!}{(n-k)!}

Example: Number of ways to award gold, silver, bronze to 10 athletes:

P(10,3) = 10 \times 9 \times 8 = 720

Permutations with Repetition

n^k

— choose from

n

options,

k

times, with repetition allowed. Example: Number of 4-digit PINs:

10^4 = 10000

1.4 Combinations

Combination: An unordered selection of distinct objects.

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

Example: Number of ways to choose a 3-person committee from 10:

\binom{10}{3} = 120

Relationship

P(n,k) = C(n,k) \times k!

1.5 Binomial Coefficients

(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k

Pascal's Identity

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

📊 Formula Summary

Concept	Formula
Permutations	$P(n,k) = \frac{n!}{(n-k)!}$
Combinations	$C(n,k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$
Permutations with repetition	$n^k$
Binomial theorem	$(x+y)^n = \sum \binom{n}{k} x^{n-k} y^k$

✅ Practice Questions

Q1: How many ways to arrange the letters in "MATH"?

Solution

4 distinct letters: $4! = 24$ arrangements. Q2: How many 5-card poker hands from a 52-card deck? Solution

$\binom{52}{5} = \frac{52!}{5!47!} = 2,598,960$ . Q3: How many ways to choose a president, VP, and secretary from 12 people? Solution

$P(12,3) = 12 \times 11 \times 10 = 1320$ . Q4: Prove $C(n,k) = C(n, n-k)$ combinatorially. Solution

Choosing $k$ elements to include is equivalent to choosing $n-k$ elements to exclude. So $\binom{n}{k} = \binom{n}{n-k}$ . Q5: How many bit strings of length 8 contain exactly three 1s? Solution

Choose 3 positions out of 8 for the 1s: $\binom{8}{3} = 56$ . Join Discord NextAdvanced Counting