Combinatorics

Rule of Sum. If task $A$ can be done in $m$ ways and task $B$ in $n$ ways, and they cannot both be Done, then $A$ or $B$ can be done in $m + n$ ways.

Rule of Product. If task $A$ can be done in $m$ ways and task $B$ in $n$ ways independently, then $A$ and $B$ together can be done in $mn$ ways.

Permutations: $P(n, r) = n! / (n-r)!$ — ordered arrangements of $r$ items from $n$.

Combinations: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ — unordered selections of $r$ items from $n$.

Theorem 4.1 (Binomial Theorem).

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

Theorem 4.2 (Pascal”s Identity). $\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$

Proof. Every $r$-subset of $\\{1, \ldots, n\\}$ either contains $n$ (giving $\binom{n-1}{r-1}$ ways To choose the remaining $r-1$) or does not contain $n$ (giving $\binom{n-1}{r}$ ways to choose all $r$ From $\\{1, \ldots, n-1\\}$). $\blacksquare$

Theorem 4.3 (Inclusion-Exclusion). For finite sets $A_1, \ldots, A_n$:

$\left|\bigcup_{i=1}^{n} A_i\right| = \sum_i |A_i| - \sum_{i \lt j} |A_i \cap A_j| + \sum_{i \lt j \lt k} |A_i \cap A_j \cap A_k| - \cdots + (-1)^{n+1}|A_1 \cap \cdots \cap A_n|$

Proof (for two sets). Every element of $A_1 \cup A_2$ is in $A_1$ or $A_2$ or both. Counting $|A_1| + |A_2|$ counts elements in $A_1 \cap A_2$ twice, so we subtract $|A_1 \cap A_2|$ once: $|A_1 \cup A_2| = |A_1| + |A_2| - |A_1 \cap A_2|$.

For the general case, an element in exactly $t$ of the sets is counted $\binom{t}{1} - \binom{t}{2} + \binom{t}{3} - \cdots = 1 - (1-1)^t = 1$ time, which is correct. $\blacksquare$

Worked Example. How many integers from 1 to 1000 are not divisible by 2, 3, or 5?

Let $A_2$ = multiples of 2, $A_3$ = multiples of 3, $A_5$ = multiples of 5.

$|A_2| = 500$, $|A_3| = 333$, $|A_5| = 200$.

$|A_2 \cap A_3| = 166$, $|A_2 \cap A_5| = 100$, $|A_3 \cap A_5| = 66$.

$|A_2 \cap A_3 \cap A_5| = 33$.

$|A_2 \cup A_3 \cup A_5| = 500 + 333 + 200 - 166 - 100 - 66 + 33 = 734$.

Not divisible by 2, 3, or 5: $1000 - 734 = 266$. $\blacksquare$

Worked Example. How many integers from 1 to 500 are divisible by 3 or 7 but not by 5?

Theorem 4.4. The number of ways to distribute $n$ identical objects into $k$ distinct bins is $\binom{n + k - 1}{k - 1}$.

Proof. The problem is equivalent to placing $k - 1$ dividers among $n$ objects, giving $\binom{n + k - 1}{n} = \binom{n + k - 1}{k - 1}$ arrangements. $\blacksquare$

Worked Example. How many solutions does $x_1 + x_2 + x_3 = 15$ have with $x_i \geq 1$?

Worked Example. How many solutions does $x_1 + x_2 + x_3 + x_4 = 20$ have with $x_i \geq 0$?

Theorem 4.5 (Pigeonhole Principle). If $n$ objects are placed into $k$ boxes and $n \gt k$Then at Least one box contains at least $\lceil n/k \rceil$ objects.

Proof. If every box contained at most $\lceil n/k \rceil - 1$ objects, the total would be at most $k(\lceil n/k \rceil - 1) \lt k \cdot n/k = n$Contradicting that there are $n$ objects. $\blacksquare$

Worked Example. In a class of 400 students, at least how many were born in the same month?

Worked Example. Show that among any $n + 1$ integers from $\\{1, 2, \ldots, 2n\\}$Two of them Differ by exactly $n$.

Worked Example. Prove that any sequence of $n^2 + 1$ distinct real numbers contains a monotone (increasing or decreasing) subsequence of length $n + 1$.

Theorem 4.6 (Generalised Pigeonhole Principle). If $n$ objects are placed into $k$ boxes, then at Least one box contains at least $\lceil n/k \rceil$ objects. Equivalently, if each box contains at most $m$ objects, then the total number of objects is at most $km$.

Worked Example. A drawer contains red, blue, and yellow socks. How many socks must be drawn (without looking) to guarantee at least 4 socks of the same colour?

Worked Example. Prove that in any group of $n$ people, there are at least two who have shaken Hands with the same number of people (within the group).

The $n$-th Catalan number is

$C_n = \frac{1}{n+1}\binom{2n}{n} = \frac{(2n)!}{(n+1)!\,n!}$

The first few values: $C_0 = 1$, $C_1 = 1$, $C_2 = 2$, $C_3 = 5$, $C_4 = 14$, $C_5 = 42$.

Catalan numbers count:

• The number of valid (properly matched) sequences of $n$ pairs of parentheses.
• The number of binary search trees with $n$ nodes.
• The number of ways to triangulate a convex $(n+2)$-gon.
• The number of lattice paths from $(0,0)$ to $(n,n)$ that never go above the diagonal.

Recurrence. $C_0 = 1$ and for $n \geq 1$:

$C_n = \sum_{i=0}^{n-1} C_i \, C_{n-1-i}$

Worked Example. Verify $C_3 = 5$ by listing all valid sequences of 3 pairs of parentheses.

The ordinary generating function (OGF) of a sequence $\\{a_n\\}$ is

$G(x) = \sum_{n=0}^{\infty} a_n x^n$

Common generating functions:

Key operations. If $A(x)$ generates $\\{a_n\\}$ and $B(x)$ generates $\\{b_n\\}$:

• $A(x) + B(x)$ generates $\\{a_n + b_n\\}$ (choice between types).
• $A(x) \cdot B(x)$ generates $\\{c_n\\}$ where $c_n = \sum_{i=0}^{n} a_i b_{n-i}$ (combining two choices).

Worked Example. Find the number of ways to select $n$ coins from unlimited supplies of 1p, 2p, And 5p coins.