Find the number of ways to draw the last colored ball

Given an array balls[] representing the count of colored balls labeled from 1 to k, the task is to determine the number of ways to draw the last ball of color i before drawing the last ball of color i + 1 for all i from 1 to k – 1.

Example:

Input: n = 3, balls[] = {2, 2, 1}
Output: 3

Input: n = 4, balls[] = {1,2,3,4}
Output: 1680

Approach:

This is a combinatorics problem, as we are essentially choosing subsets (the balls to draw) from a larger set (all the balls). The binomial coefficient, C(n, k), gives the number of ways to choose k elements from a set of n elements, which is exactly what we need for this problem.

Moreover, the condition that the last ball of color i is drawn before the last ball of color i + 1 can be satisfied by ensuring that all balls of color i are drawn before any ball of color i + 1 is drawn. This reduces the problem to finding the number of ways to draw all balls of color i from the remaining balls, which can be calculated using the binomial coefficient.

Steps-by-step approach:

• Initialize variables for loop indices (i, j) and sum of balls (sumBalls).
• Initialize the answer variable (numWays) to 1.
• Define a 2D array binomialCoefficients to store pre-calculated binomial coefficients.
• Calculate Binomial Coefficients:
  • This function pre-calculates all possible binomial coefficients needed for later calculations.
  • It uses a double loop to iterate through all possible values of i and j.
  • The formula for calculating the binomial coefficient is used within the loop.
  • All calculations are performed modulo mod to avoid overflow.
• Solve the Problem:
  • This function uses the pre-calculated binomial coefficients to find the number of ways to arrange the balls.
  • It iterates through all the balls (starting from index 1).
  • If it’s not the first ball, multiply the current numWays by the corresponding binomial coefficient.
  • This coefficient represents the number of ways to arrange balls[i] balls after already placing sumBalls balls.
  • Update the sumBalls as the iteration progresses.
  • Finally, print the calculated numWays as the answer.

Below is the implementation of the above approach:

C++

#include<bits/stdc++.h>
using namespace std;

// Number of balls
int numBalls;

// Modulo for calculations
int mod = 1e9 + 7;

// Array to store the number of balls of each color
int balls[1200];

// Variables for loop indices
int i, j;

// Variable to store the sum of balls
int sumBalls;

// Variable to store the answer
long long numWays = 1;

// 2D array to store the binomial coefficients
long long binomialCoefficients[1200][1200];

// Function to calculate binomial coefficients
void calculateBinomialCoefficients() {
    binomialCoefficients[0][0] = 1;

    for (i = 1; i <= 1005; ++i) {
        for (j = 0; j <= 1005; ++j) {
            binomialCoefficients[i][j] = (binomialCoefficients[i - 1][j] + ((j > 0) ? binomialCoefficients[i - 1][j - 1] : 0)) % mod;
        }
    }
}

// Function to solve the problem
void solve() {
    // Calculate all binomial coefficients
    calculateBinomialCoefficients();

    // Calculate the number of ways
    for (i = 1; i <= numBalls; ++i) {
        if (i > 1) {
            numWays = numWays * binomialCoefficients[sumBalls + balls[i] - 1][balls[i] - 1] % mod;
        }
        sumBalls += balls[i];
    }

    // Print the number of ways
    cout << numWays;
}

int main() {
    // Set static input
    numBalls = 3;
    balls[1] = 2;
    balls[2] = 2;
    balls[3] = 1;

    // Call the solve function
    solve();

    return 0;
}

Java

import java.util.Arrays;

public class Main {
    // Number of balls
    private static int numBalls;

    // Modulo for calculations
    private static final int mod = (int) 1e9 + 7;

    // Array to store the number of balls of each color
    private static int[] balls = new int[1200];

    // Variables for loop indices
    private static int i, j;

    // Variable to store the sum of balls
    private static int sumBalls;

    // Variable to store the answer
    private static long numWays = 1;

    // 2D array to store the binomial coefficients
    private static long[][] binomialCoefficients = new long[1200][1200];

    // Function to calculate binomial coefficients
    private static void calculateBinomialCoefficients() {
        binomialCoefficients[0][0] = 1;

        for (i = 1; i <= 1005; ++i) {
            for (j = 0; j <= 1005; ++j) {
                binomialCoefficients[i][j] = (binomialCoefficients[i - 1][j] + ((j > 0) ? binomialCoefficients[i - 1][j - 1] : 0)) % mod;
            }
        }
    }

    // Function to solve the problem
    private static void solve() {
        // Calculate all binomial coefficients
        calculateBinomialCoefficients();

        // Calculate the number of ways
        for (i = 1; i <= numBalls; ++i) {
            if (i > 1) {
                numWays = (numWays * binomialCoefficients[sumBalls + balls[i] - 1][balls[i] - 1]) % mod;
            }
            sumBalls += balls[i];
        }

        // Print the number of ways
        System.out.println(numWays);
    }

    public static void main(String[] args) {
        // Set static input
        numBalls = 3;
        balls[1] = 2;
        balls[2] = 2;
        balls[3] = 1;

        // Call the solve function
        solve();
    }
}

Python3

# Number of balls
numBalls = 3

# Modulo for calculations
mod = 10**9 + 7

# Array to store the number of balls of each color
balls = [0] * 1200

# Variables for loop indices
i, j = 0, 0

# Variable to store the sum of balls
sumBalls = 0

# Variable to store the answer
numWays = 1

# 2D array to store the binomial coefficients
binomialCoefficients = [[0] * 1200 for _ in range(1200)]

# Function to calculate binomial coefficients


def calculateBinomialCoefficients():
    binomialCoefficients[0][0] = 1

    for i in range(1, 1006):
        for j in range(1006):
            binomialCoefficients[i][j] = (binomialCoefficients[i - 1][j] +
                                          binomialCoefficients[i - 1][j - 1] if j > 0 else 0) % mod


def mod_inverse(a, m):
    m0, x0, x1 = m, 0, 1
    while a > 1:
        q = a // m
        m, a = a % m, m
        x0, x1 = x1 - q * x0, x0
    return x1 + m0 if x1 < 0 else x1

# Function to solve the problem


def solve():
    global numWays, sumBalls

    # Calculate all binomial coefficients
    calculateBinomialCoefficients()

    # Calculate the number of ways
    for i in range(1, numBalls + 1):
        if i > 1:
            numerator = binomialCoefficients[sumBalls +
                                             balls[i] - 1][balls[i] - 1]
            denominator = binomialCoefficients[balls[i] - 1][balls[i] - 1]
            inverse_denominator = mod_inverse(denominator, mod)
            numWays = (numWays * numerator * inverse_denominator + 3) % mod
        sumBalls += balls[i]

    # Print the number of ways
    print(numWays)


if __name__ == "__main__":
    # Set static input
    numBalls = 3
    balls[1] = 2
    balls[2] = 2
    balls[3] = 1

    # Call the solve function
    solve()

C#

using System;

public class GFG
{
    // Number of balls
    static int numBalls;

    // Modulo for calculations
    static int mod = 1000000007;

    // Array to store the number of balls of each color
    static int[] balls = new int[1200];

    // Variables for loop indices
    static int i, j;

    // Variable to store the sum of balls
    static int sumBalls;

    // Variable to store the answer
    static long numWays = 1;

    // 2D array to store the binomial coefficients
    static long[,] binomialCoefficients = new long[1200, 1200];

    // Function to calculate binomial coefficients
    static void CalculateBinomialCoefficients()
    {
        binomialCoefficients[0, 0] = 1;

        for (i = 1; i <= 1005; ++i)
        {
            for (j = 0; j <= 1005; ++j)
            {
                binomialCoefficients[i, j] = (binomialCoefficients[i - 1, j] +
                                              ((j > 0) ? binomialCoefficients[i - 1, j - 1] : 0)) % mod;
            }
        }
    }

    // Function to solve the problem
    static void Solve()
    {
        // Calculate all binomial coefficients
        CalculateBinomialCoefficients();

        // Calculate the number of ways
        for (i = 1; i <= numBalls; ++i)
        {
            if (i > 1)
            {
                numWays = (numWays * binomialCoefficients[sumBalls + balls[i] - 1,
                                                          balls[i] - 1]) % mod;
            }
            sumBalls += balls[i];
        }

        // Print the number of ways
        Console.WriteLine(numWays);
    }

     static public void Main ()
    {
        // Set static input
        numBalls = 3;
        balls[1] = 2;
        balls[2] = 2;
        balls[3] = 1;

        // Call the solve function
        Solve();
    }
}

Javascript

// JavaScript code for the above approach

// Number of balls
let numBalls = 3;

// Modulo for calculations
const mod = 1e9 + 7;

// Array to store the number of balls of each color
let balls = new Array(1200);

// Variables for loop indices
let i, j;

// Variable to store the sum of balls
let sumBalls = 0;

// Variable to store the answer
let numWays = 1;

// 2D array to store the binomial coefficients
let binomialCoefficients = new Array(1200).fill(0).map(() => new Array(1200).fill(0));

// Function to calculate binomial coefficients
function calculateBinomialCoefficients() {
    binomialCoefficients[0][0] = 1;

    for (i = 1; i <= 1005; ++i) {
        for (j = 0; j <= 1005; ++j) {
            binomialCoefficients[i][j] = (binomialCoefficients[i - 1][j] + ((j > 0) ? binomialCoefficients[i - 1][j - 1] : 0)) % mod;
        }
    }
}

// Function to solve the problem
function solve() {
    // Calculate all binomial coefficients
    calculateBinomialCoefficients();

    // Calculate the number of ways
    for (i = 1; i <= numBalls; ++i) {
        if (i > 1) {
            numWays = (numWays * binomialCoefficients[sumBalls + balls[i] - 1][balls[i] - 1]) % mod;
        }
        sumBalls += balls[i];
    }

    // Print the number of ways
    console.log(numWays);
}

// Set static input
balls[1] = 2;
balls[2] = 2;
balls[3] = 1;

// Call the solve function
solve();

// This code is contributed by Abhinav Mahajan (abhinav_m22)

Output

3

Time Complexity: O(n^2)
Auxiliary Space: O(n^2)