Unit 6: Permutations, Combinations &
Binomial Theorem

Factorials & Basic Facts

• Definition: \( n! = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1 \), for \( n \geq 1 \).
• Convention: \( 0! = 1 \).
• Fast cancellations: \( \frac{(n+2)!}{n!} = (n+2)(n+1) \), \( \frac{n!}{(n-2)!} = n(n-1) \).

Counting Principle

• If Task A has \( m \) ways and for each, Task B has \( n \) ways, then A then B has \( m \cdot n \) ways.
• Extend by multiplication for more steps.

Permutations vs Combinations

Useful Combinatorial Identities

• \( C(n,k) = C(n,n-k) \)
• Pascal: \( C(n,k) = C(n-1,k-1) + C(n-1,k) \)
• \( P(n,k) = C(n,k) \cdot k! \)

Quick Reference: Factorial Simplifications

• \( \frac{(n+a)!}{n!} = (n+a)(n+a-1)\ldots(n+1) \)
• Cancel common factorial factors before expanding to simplify algebraic solving.

Arranging People/Objects in a Line

• No restrictions: \( n! \)
• Two together: \( (n-1)! \cdot 2 \)
• Three together: \( (n-2)! \cdot 3! \)
• Two NOT together: \( n! - (n-1)! \cdot 2 \)
• Alternating boys/girls (equal m): \( 2 \cdot m! \cdot m! \)
• Boy–girl couples (fixed): \( 5! \cdot 2^5 = 3,840 \)
• Boy–girl couples (any pairing): \( 5! \cdot 5! \cdot 2^5 = 460,800 \)

Block & Complement Techniques

• Block method: group required-adjacent persons as one “super-person”; multiply by internal arrangements.
• Complement method: count total arrangements then subtract arrangements that violate the restriction.

Solving Factorial, P, C Expressions

• Evaluate \( P \) and \( C \) by canceling factorials first.
• Examples: \( P(7,3)=210 \), \( C(8,3)=56 \).

Solving Equations with Factorials, P, C

• Reduce factorial ratios to a polynomial in \( n \); solve integer solutions.
• Example: \( C(n,2)=15 \Rightarrow n=6 \).
• Example: \( P(n,4)=840 \Rightarrow n=7 \).
• Example: \( \frac{(n+2)!}{n!}=56 \Rightarrow n=6 \).

Binomial Coefficients & Pascal’s Triangle

• Row \( n \) gives coefficients \( C(n,0), C(n,1), \ldots, C(n,n) \).
• Coefficients for \( (a+b)^n \) are row \( n \) of Pascal's triangle.

Binomial Theorem — Expansion Formula

• \( (a+b)^n = \sum_{k=0}^n C(n,k) a^{n-k} b^k \)
• General term: \( T_{k+1} = C(n,k) a^{n-k} b^k \)
• First term: \( k=0 \Rightarrow a^n \). Last term: \( k=n \Rightarrow b^n \).

Finding Particular Terms

• For \( (x+1)^n \): \( T_{k+1} = C(n,k) x^{n-k} \). Term with \( x^r \): \( k=n-r \).
• Coefficient of \( x^r \) is \( C(n, r) \).
• Constant term: \( r=0 \Rightarrow k=n \Rightarrow 1 \).
• For \( (x^p+x^q)^n \): exponent on \( x \) in \( T_{k+1} \): \( E = p(n-k) + qk \).
• For \( (x^3+x^2)^n \): \( E = 3n-k \). To find \( x^r \): \( k=3n-r \).
• Constant term: solve \( pn+k(q-p)=0 \).