Ways to Arrange Balls

Hard

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Problem statement

Given ‘a’ balls of type ‘A’, ‘b’ balls of type ‘B’ and ‘c’ balls of type ‘C’. You need to find the total number of ways to arrange the balls in a straight line such that adjacent balls are of different types.

In other words, find the total ways to arrange the given balls in such a way that no two balls of the same type are adjacent.

For Example :

Suppose we have 2 balls of type ‘A’, 1 ball of type ‘B’ and 1 ball of type ‘C’, following are the ways to arrange them in the required fashion. 
ABCA
ABAC 
ACBA 
ACAB 
BACA 
CABA 
Hence, there is a total of six ways.

Input Format :

The first line of input contains an integer ‘T’ denoting the number of test cases.

The first line of each test contains three space-separated positive integers a, b, c, denoting the number of balls of type ‘A’, ‘B’, and ‘C’ respectively.

Output Format :

For each test case, print the total ways to arrange the balls such that adjacent balls are of a different type in a single line.

Print the output of each test case in a separate line.

Note :

You don’t have to print anything, it has already been taken care of. Just complete the function. 

If there is no way, return 0.

Constraints :

1 <= T <= 100
0 <= a, b, c <= 15

Time limit: 1 sec

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Approaches

01 Approach

This problem can be solved by solving its subproblems and then combining the solutions of the solved subproblems to solve the original problem. We will do this using recursion. We will keep track of the next ball we want to add in the line. Initially, we have three options, to begin with. We can start by adding A or B or C. After that, at each step, we have two options. We can find the number of ways to make the required arrangements from the remaining balls.

Algorithm:

We will call a recursive function ‘helper’ that takes all the three integers a, b, c, and a variable ‘nextBall’ as arguments and returns us the total number of ways to arrange the balls in the required fashion.
Initialize a variable called ‘totalWays’ by 0.
Call helper function for all possible value of ‘nextBall’ and store the answer in ‘totalWays’, i.e. totalWays = helper(a, b, c, 0) + helper(a, b, c, 1) + helper(a, b, c, 2).
The algorithm for the helper function will be as follows:
Int helper(a, b, c, nextBall):
A. If ‘a’ or ‘b’ or ‘c’ becomes less than 0, return 0 as we can’t have an arrangement in this case.
B. Now check,
- if the ‘nextBall’ is 0, means we have to add a ball of type A.
  - If ‘a’ is 1 and ‘b’ and ‘c’ are 0, return 1 as we have found an arrangement.
  - Else return helper(a - 1, b, c, 1) + helper(a - 1, b, c, 2).
- If the ‘nextBall’ is 1, means we have to add a ball of type B.
  - If ‘b' is 1 and ‘a’ and ‘c’ are 0, return 1 as we have found an arrangement.
  - Else return helper(a , b - 1, c, 0) + helper(a, b - 1, c, 2).
- If the ‘nextBall’ is 2, means we have to add a ball of type C.
  - If ‘c’ is 1 and ‘a’ and ‘b’ are 0, return 1 as we have found an arrangement.
  - Else return helper(a, b, c - 1, 0) + helper(a, b, c - 1, 1).
Return the ‘totalWays’.

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02 Approach

In the recursive approach, there are many overlapping sub-problems. This problem can be solved using memoization.

Algorithm

The idea is to use memoization.
Create a 4D lookup table to store the solutions to the subproblems where lookUp[i][j][k][l] denotes the number of ways to arrange the ‘i’ balls of type A,’j’ balls of type B and ‘k’ balls of type C in the required fashion when the next ball we need to add in the line will be known from the value of ‘l’, that is when ‘l’ is 0, we need to place a ball of type ‘A’ and when ‘l’ is 1, we need to place a ball of type B and when ‘l’ is 2, we need to place a ball of type C. Note that ‘l’ can only attain three values, 0, 1, and 2.
Initialize the table by -1.
If the value is already computed, return the value.
Otherwise, compute the value using recursion and store the result in ‘lookUp’ table and return it.

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03 Approach

The idea is to use Dynamic Programming.

We will create a 4D lookup table, DP, to store the solutions to the subproblems where DP[i][j][k][l] denotes the number of ways to arrange the ‘i’ balls of type A,’j’ balls of type B and ‘k’ balls of type C in the required fashion when the next ball we need to add in the line will be known from the value of ‘l’, that is when ‘l’ is 0, we need to place a ball of type ‘A’ and when ‘l’ is 1, we need to place a ball of type B and when ‘l’ is 2, we need to place a ball of type C.

Note that ‘l’ can only attain three values, 0, 1, and 2.

Algorithm

Initialize the table by 0.
The detailed algorithm to fill the DP table will be as follows:
- Loop 1: For i = 0 to a+1:
- Loop 2: For j = 0 to b+1:
- Loop 3: For k = 0 to c+1:
- Loop 4: For l = 0 to 3:
if the ‘l’ is 0, means we have to add a ball of type A.
- If i is 1 and j and k are 0, update DP[i][j][k][l] =1 as we have found an arrangement.
- Else if i >0, update DP[i][j][k][l] += DP[i-1][j][k][1] + DP[i-1][j][k][2];
If the ‘l’ is 1, means we have to add a ball of type B.
- If j is 1 and i and k are 0, update DP[i][j][k][l] =1 as we have found an arrangement.
- Else if j > 0, update DP[i][j][k][l] += DP[i][j-1][k][1] + DP[i][j-1][k][2];
If the ‘l’ is 2, means we have to add a ball of type C.
- If k is 1 and i and j are 0, update DP[i][j][k][l] =1 as we have found an arrangement.
- Else if k > 0, update DP[i][j][k][l] += DP[i][j][k-1[1] + DP[i][j][k-1][2];
Return (DP[a][b][c][0] + DP[a][b][c][1] + DP[a][b][c][2]) as it is the total no. of ways to arrange the balls in the required manner.