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# In how many ways the ball will come back to the first boy after N turns

• Last Updated : 20 Aug, 2022

Four boys are playing a game with a ball. In each turn, the player (who has the ball currently) passes it to a different player randomly. Bob always starts the game. The task is to find in how many ways the ball will come back to Bob after N passes.
Examples:

Input: N = 3
Output:
Here are all the possible ways:
Bob -> Boy1 -> Boy2 -> Bob
Bob -> Boy1 -> Boy3 -> Bob
Bob -> Boy2 -> Boy1 -> Bob
Bob -> Boy2 -> Boy3 -> Bob
Bob -> Boy3 -> Boy1 -> Bob
Bob -> Boy3 -> Boy2 -> Bob
Input: N = 10
Output: 14763

Approach: Let the number of sequences that get back to Bob after N passes are P(N). There are two cases, either pass N – 2 is to Bob or it is not. Note that Bob can’t have the ball at (N – 1)th pass because then he won’t have the ball at the Nth pass.

1. Case 1: If pass N – 2 is to Bob then the pass N – 1 can be to any of the other 3 boys. Thus, the number of such sequences is 3 * P(N – 2).
2. Case 2: If pass N – 2 is not to Bob then pass N – 1 is to one of the 2 boys other than Bob and the one who got the ball in hand. So, substitute Bob for the receiver of pass N – 1, and get a unique N – 1 sequence. So, the number of such sequences are 2 * P(N – 1).

Hence the recurrence relation will be P(N) = 2 * P(N – 1) + 3 * P(N – 2) where P(0) = 1 and P(1) = 0.
After solving the recurrence relation, P(N) = (3N + 3 * (-1N)) / 4
Below is the implementation of the above approach:

## C++

 `// Function to return the number of` `// sequences that get back to Bob` `#include ` `using` `namespace` `std;`   `int` `numSeq(``int` `n)` `{` `    ``return` `(``pow``(3, n) + 3 * ``pow``(-1, n)) / 4;` `}`   `// Driver code` `int` `main()` `{` `    ``int` `N = 10;` `    ``printf``(``"%d"``, numSeq(N));` `    ``return` `0;` `}`   `// This code is contributed by Mohit kumar`

## Java

 `// Function to return the number of` `// sequences that get back to Bob` `import` `java.util.*;`   `class` `GFG` `{`   `static` `int` `numSeq(``int` `n)` `{` `    ``return` `(``int``) ((Math.pow(``3``, n) + ``3` `* ` `                    ``Math.pow(-``1``, n)) / ``4``);` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `N = ``10``;` `    ``System.out.printf(``"%d"``, numSeq(N));` `}` `}`   `// This code is contributed by Rajput-Ji`

## Python3

 `# Function to return the number of ` `# sequences that get back to Bob` `def` `numSeq(n):` `    ``return` `(``pow``(``3``, n) ``+` `3` `*` `pow``(``-``1``, n))``/``/``4` `    `  `# Driver code ` `N ``=` `10` `print``(numSeq(N))`

## C#

 `// C# implementation of the above approach` `using` `System;`   `// Function to return the number of ` `// sequences that get back to Bob ` `class` `GFG ` `{ `   `    ``static` `int` `numSeq(``int` `n) ` `    ``{ ` `        ``return` `(``int``) ((Math.Pow(3, n) + 3 * ` `                       ``Math.Pow(-1, n)) / 4); ` `    ``} ` `    `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `N = 10; ` `        ``Console.WriteLine(numSeq(N)); ` `    ``} ` `} `   `// This code is contributed by AnkitRai01`

## Javascript

 ``

Output:

`14763`

Time Complexity: O(log n)

Auxiliary Space: O(1)

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