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# Combinatorics

Authors: Jesse Choe, Aadit Ambadkar, Dustin Miao

How to count.

### Prerequisites

If you've never encountered any combinatorics before, AoPS is a good place to start.

Resources
AoPS

practice problems, set focus to Counting and Probability.

AoPS

good book

## Resources

Resources
CPH

module is based off of this

cp-algo
HE

teaches fundamental combinatorics with a practice problem at the end

AoPS

teaches basic combinatorics concepts

AoPS

teaches more advanced combinatorics concepts

CF

a good blog about the expected value

CF

a good blog about the inclusion-exclusion principle

If you prefer watching videos instead, here are some options:

Resources

playlist by mathemaniac

lectures 16-23

Errichto video regarding expected value and sums of subsets

## Binomial Coefficients

Focus Problem – try your best to solve this problem before continuing!

The binomial coefficient $\binom{n}{k}$ (pronounced as "$n$ choose $k$" or sometimes written as ${}_nC_k$) represents the number of ways to choose a subset of $k$ elements from a set of $n$ elements. For example, $\binom{4}{2} = 6$, because the set $\{1,2,3,4\}$ has $6$ subsets of $2$ elements:

$\{1, 2\}, \{1, 3\}, \{1, 4\}, \{2, 3\}, \{2, 4\}, \{3, 4\}$

There are two ways to calculate binomial coefficients:

### Method 1: Pascal's Triangle (Dynamic Programming) - $\mathcal{O}(n^2)$

Binomial coefficients can be recursively calculated as follows:

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

The intuition behind this is to fix an element $x$ in the set and choose $k − 1$ elements from $n − 1$ elements if $x$ is included in the set or choose $k$ elements from $n − 1$ elements, otherwise.

The base cases for the recursion are:

$\binom{n}{0} = \binom{n}{n} = 1$

because there is always exactly one way to construct an empty subset and a subset that contains all the elements.

This recursive formula is commonly known as Pascal's Triangle.

A naive implementation of this would use a recursive formula, like below:

C++

/** @return nCk mod p using naive recursion */int binomial(int n, int k, int p) {	if (k == 0 || k == n) { return 1; }	return (binomial(n - 1, k - 1, p) + binomial(n - 1, k, p)) % p;}

Java

/** @return nCk mod p using naive recursion */public static int binomial(int n, int k, int p) {	if (k == 0 || k == n) { return 1; }	return (binomial(n - 1, k - 1, p) + binomial(n - 1, k, p)) % p;}

Python

def binomial(n: int, k: int, p: int) -> int:	""":return: nCk mod p using naive recursion"""	if k == 0 or k == n:		return 1	return (binomial(n - 1, k - 1, p) + binomial(n - 1, k, p)) % p

Additionally, we can optimize this from $\mathcal{O}(2^n)$ to $\mathcal{O}(n^2)$ using dynamic programming (DP) by caching the values of smaller binomials to prevent recalculating the same values over and over again. The code below shows a bottom-up implementation of this.

C++

/** @return nCk mod p using dynamic programming */int binomial(int n, int k, int p) {	// dp[i][j] stores iCj	vector<vector<int>> dp(n + 1, vector<int>(k + 1, 0));
// base cases described above	for (int i = 0; i <= n; i++) {		/*		 * i choose 0 is always 1 since there is exactly one way		 * to choose 0 elements from a set of i elements

Java

/** @return nCk mod p using dynamic programming */public static int binomial(int n, int k, int p) {	// dp[i][j] stores iCj	int[][] dp = new int[n + 1][k + 1];
// base cases described above	for (int i = 0; i <= n; i++) {		/*		 * i choose 0 is always 1 since there is exactly one way		 * to choose 0 elements from a set of i elements

Python

def binomial(n: int, k, p):	""":return: nCk mod p using dynamic programming"""	# dp[i][j] stores iCj	dp = [[0] * (k + 1) for _ in range(n + 1)]
# base cases described above	for i in range(n + 1):		"""		i choose 0 is always 1 since there is exactly one way		to choose 0 elements from a set of i elements

### Method 2: Factorial Definition (Modular Inverses) - $\mathcal{O}(n + \log MOD)$

Define $n!$ as $n \times (n - 1) \times (n - 2) \times \ldots 1$. $n!$ represents the number of permutations of a set of $n$ elements. See this AoPS Article for more details.

Another way to calculate binomial coefficients is as follows:

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

Recall that $\binom{n}{k}$ also represents the number of ways to choose $k$ elements from a set of $n$ elements. One strategy to get all such combinations is to go through all possible permutations of the $n$ elements, and only pick the first $k$ elements out of each permutation. There are $n!$ ways to do so. However, note the the order of the elements inside and outside the subset does not matter, so the result is divided by $k!$ and $(n − k)!$.

Since these binomial coefficients are large, problems typically require us to output the answer modulo a large prime $p$ such as $10^9 + 7$. Fortunately, we can use modular inverses to divide $n!$ by $k!$ and $(n - k)!$ modulo $p$ for any prime $p$. Computing inverse factorials online can be very time costly. Instead, we can precompute all factorials in $\mathcal{O}(n)$ time and inverse factorials in $\mathcal{O}(n + \log MOD)$. First, we compute the modular inverse of the largest factorial using binary exponentiation. For the rest, we use the fact that $(n!)^{-1} \equiv (n!)^{-1}\times (n+1)^{-1} \times (n+1) \equiv ((n+1)!)^{-1}\times (n+1)$. See the code below for the implementation.

C++

const int MAXN = 1e6;
long long fac[MAXN + 1];long long inv[MAXN + 1];
/** @return x^n modulo m in O(log p) time. */long long exp(long long x, long long n, long long m) {	x %= m;  // note: m * m must be less than 2^63 to avoid ll overflow	long long res = 1;	while (n > 0) {

Java

import java.util.*;
public class BinomialCoefficients {	private static final int MAXN = (int)1e6;	private static long[] fac = new long[MAXN + 1];	private static long[] inv = new long[MAXN + 1];
/** @return x^n modulo m in O(log p) time. */	private static long exp(long x, long n, long m) {		x %= m;  // note: m * m must be less than 2^63 to avoid ll overflow

Python

MAXN = 10**6
fac = [0] * (MAXN + 1)inv = [0] * (MAXN + 1)

def exp(x: int, n: int, m: int) -> int:	""":return: x^n modulo m in O(log p) time."""	x %= m  # note: m * m must be less than 2^63 to avoid ll overflow	res = 1

### Solution - Binomial Coefficients

The first method for calculating binomial factorials is too slow for this problem since the constraints on $a$ and $b$ are $(1 \leq b \leq a \leq 10^6)$ (recall that the first implementation runs in $\mathcal{O}(n^2)$ time complexity). However, we can use the second method to answer each of the $n$ queries in constant time by precomputing factorials and their modular inverses.

C++

#include <iostream>using namespace std;using ll = long long;
const int MAXN = 1e6;const int MOD = 1e9 + 7;
ll fac[MAXN + 1];ll inv[MAXN + 1];


Java

import java.io.*;import java.util.*;
public class BinomialCoefficients {	private static final int MAXN = (int)1e6;	private static final int MOD = (int)1e9 + 7;	private static long[] fac = new long[MAXN + 1];	private static long[] inv = new long[MAXN + 1];
public static void main(String[] args) {

Python

MAXN = 10**6MOD = 10**9 + 7
fac = [0] * (MAXN + 1)inv = [0] * (MAXN + 1)

Code Snippet: Counting Functions (Click to expand)



## Derangements

Focus Problem – try your best to solve this problem before continuing!

The number of derangements of $n$ numbers, expressed as $!n$, is the number of permutations such that no element appears in its original position. Informally, it is the number of ways $n$ hats can be returned to $n$ people such that no person recieves their own hat.

### Method 1: Principle of Inclusion-Exclusion

Suppose we had events $E_1, E_2, \dots, E_n$, where event $E_i$ corresponds to person $i$ recieving their own hat. We would like to calculate $n! - \lvert E_1 \cup E_2 \cup \dots \cup E_n \rvert$.

We subtract from $n!$ the number of ways for each event to occur; that is, consider the quantity $n! - \lvert E_1 \rvert - \lvert E_2 \rvert - \dots - \lvert E_n \rvert$. This undercounts, as we are subtracting cases where more than one event occurs too many times. Specifically, for a permutation where at least two events occur, we undercount by one. Thus, add back the number of ways for two events to occur. We can continue this process for every size of subsets of indices. The expression is now of the form:

$n! - \lvert E_1 \cup E_2 \cup \dots \cup E_n \rvert = \sum_{k = 1}^n (-1)^k \cdot (\text{number of permutations with k fixed points})$

For a set size of $k$, the number of permutations with at least $k$ indicies can be computed by choosing a set of size $k$ that are fixed, and permuting the other indices. In mathematical terms:

${n \choose k}(n-k)! = \frac{n!}{k!(n-k)!}(n-k)! = \frac{n!}{k!}$

Thus, the problem now becomes computing

$n!\sum_{k=0}^n\frac{(-1)^k}{k!}$

which can be done in linear time.

C++

#include <atcoder/modint>#include <bits/stdc++.h>
using namespace std;using mint = atcoder::modint;
int main() {	int n, m;	cin >> n >> m;	mint::set_mod(m);

Java

import java.util.Scanner;
public class Main {	public static void main(String[] args) {		Scanner scanner = new Scanner(System.in);		int n = scanner.nextInt();		int m = scanner.nextInt();
long c = 1;		for (int i = 1; i <= n; i++) {

Python

n, m = map(int, input().split())c = 1for i in range(1, n + 1):	c = (c * i) + (-1 if i % 2 == 1 else 1)	c %= m	c += m	c %= m	print(c, end=" ")
print()

### Method 2: Dynamic Programming

Suppose person 1 recieved person $i$'s hat. There are two cases:

1. If person $i$ recieves person 1's hat, then the problem is reduced to a subproblem of size $n - 2$. There are $n - 1$ possibilities for $i$ in this case, so we add to the current answer $(n - 1)\cdot !(n - 2)$.
2. If person $i$ does not recieve person 1's hat, then we can reassign person 1's hat to be person $i$'s hat (if they recieved person 1's hat, then this would become first case). Thus, this becomes a subproblems with size $n - 1$, are there $n - 1$ ways to choose $i$.

Thus, we have

$!n = (n - 1)(!(n - 2) + !(n - 1))$

which can be computed in linear time with a simple DP. The base cases are that $!0 = 1$ and $!1 = 0$.

C++

#include <atcoder/modint>#include <bits/stdc++.h>
using namespace std;using mint = atcoder::modint;
int main() {	int n, m;	cin >> n >> m;	mint::set_mod(m);

Java

import java.util.Scanner;
public class Main {	public static void main(String[] args) {		Scanner scanner = new Scanner(System.in);		int n = scanner.nextInt();		int m = scanner.nextInt();
int a = 1;		int b = 0;

Python

n, m = map(int, input().split())a, b = 1, 0
print(0, end=" ")for i in range(2, n + 1):	c = (i - 1) * (a + b) % m	print(c, end=" ")	a, b = b, c
print()

## Problems

StatusSourceProblem NameDifficultyTags
CSESEasy
Show TagsCombinatorics
CSESEasy
Show TagsCombinatorics
CFEasy
Show TagsCombinatorics
CFEasy
Show TagsBinary Search, Combinatorics
BronzeEasy
Show TagsCombinatorics
Bubble CupNormal
Show TagsCombinatorics
CFNormal
Show TagsCombinatorics
ACNormal
Show TagsCombinatorics
GoldNormal
Show TagsCombinatorics
GoldNormal
Show TagsBitset, PIE
CFNormal
Show TagsCombinatorics, DP
GoldNormal
Show TagsCombinatorics, Prefix Sums
CFHard
Show TagsCombinatorics, DP
GoldInsane
Show TagsBinary Search, Combinatorics, Math, Probability

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