Question 1175169: For the upcoming world-cup, the Indian Cricket Selection Committee has to come up with a possible batting order for their players. Instead of using the traditional approach they have decided to use computer algorithms to come up with all the possible batting orders and then decide from that. The
algorithm however requires the possible batting positions for each player.
The algorithm takes a list of 11 players. Each player can have more than one position they can bat at. Your job for now is to help the selection committee calculate the total number of unique batting charts such that every player gets exactly one batting position from their list of positions and no two players are given the same batting position in one batting chart.
Player / < position 1> / < position 2> / < position 3>….
Ex:
P1 / 1 / 2 / 3 / 4
P2 / 1 / 5 / 9 / 2 / 6 / 7 / 8
P3 / 1 / 2 / 7 / 10 / 3
P4 / 1 / 9 / 2 / 6 / 7 / 10 / 3 / 4
P5 / 5 / 9 / 2 / 8 / 3 / 4
P6 / 1 / 5 / 3 / 6
P7 / 6 / 7 / 4
P8 / 1 / 9 / 2 / 4
P9 / 9 / 6 / 11 / 3 / 4
P10 / 1 / 5 / 9 / 7 / 8 / 4
P11 / 6 / 11 / 7 / 10
The total number of allocations possible is: 4646.
How to arrive at this solution?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! This problem is a classic example of a combinatorial problem that requires careful consideration of constraints and efficient algorithms. Here's how to approach it and arrive at the solution.
**Understanding the Problem**
* We have 11 players and 11 batting positions.
* Each player has a list of possible batting positions.
* We need to find the number of unique batting orders where:
* Each player is assigned exactly one position.
* No two players are assigned the same position.
**Approach: Backtracking and Constraint Satisfaction**
The most effective way to solve this is using a backtracking algorithm. Backtracking is a problem-solving technique that explores all possible solutions incrementally, abandoning partial solutions that violate constraints.
Here's a breakdown of the algorithm:
1. **Represent the Data:**
* Store the player's possible positions in a data structure (e.g., a list of lists or a dictionary).
2. **Backtracking Function:**
* Create a recursive function that takes the current player index and the current batting order as input.
* **Base Case:** If all 11 players have been assigned positions, increment a counter for the number of valid batting orders.
* **Recursive Step:**
* For the current player, iterate through their possible positions.
* For each possible position:
* Check if the position is already assigned to another player.
* If the position is available, assign it to the current player.
* Recursively call the function for the next player.
* After the recursive call, unassign the position (backtrack) to explore other possibilities.
3. **Constraints:**
* The main constraint is that no two players can have the same batting position.
**Python Code Example**
```python
def count_batting_orders(player_positions):
"""Counts the number of valid batting orders."""
num_players = len(player_positions)
assigned_positions = [0] * num_players
count = 0
def backtrack(player_index):
nonlocal count
if player_index == num_players:
count += 1
return
for position in player_positions[player_index]:
if position not in assigned_positions[:player_index]:
assigned_positions[player_index] = position
backtrack(player_index + 1)
assigned_positions[player_index] = 0 # Backtrack
backtrack(0)
return count
# Example input
player_positions = [
[1, 2, 3, 4],
[1, 5, 9, 2, 6, 7, 8],
[1, 2, 7, 10, 3],
[1, 9, 2, 6, 7, 10, 3, 4],
[5, 9, 2, 8, 3, 4],
[1, 5, 3, 6],
[6, 7, 4],
[1, 9, 2, 4],
[9, 6, 11, 3, 4],
[1, 5, 9, 7, 8, 4],
[6, 11, 7, 10],
]
result = count_batting_orders(player_positions)
print("Total number of allocations possible:", result)
```
**Explanation of the Code**
* The `backtrack` function explores all possible assignments.
* The `assigned_positions` array keeps track of what positions have been assigned.
* The `nonlocal count` line allows the inner function to modify the count variable of the outer function.
* The code iterates through the positions available to each player.
* The code checks if the position is already in the `assigned_positions` array.
* When all players have been assigned, the count is incremented.
* The code backtracks by setting the assigned position to 0, so that other options can be explored.
**Why This Approach Works**
* **Exhaustive Search:** Backtracking systematically explores all possible combinations.
* **Constraint Enforcement:** The checks ensure that the constraints are always met.
* **Efficiency:** While backtracking can be computationally expensive for very large problems, it's efficient enough for this scenario.
**Key Optimization Considerations**
* **Ordering of Positions:** If you sort the possible positions for each player, you might be able to prune the search tree earlier.
* **Heuristics:** You could use heuristics to guide the search, such as assigning players with fewer possible positions first.
By implementing this backtracking algorithm, you'll be able to calculate the 4646 possible batting orders.
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
As I read the problem and the solution by @CPhill, my impression is as follows.
(1) The problem has no any relation to Math.
(2) It teaches nothing that relates to Math.
(3) The major goal why it was posted to this forum is two-fold:
(a) to show how smart is the author,
and
(b) to show how smart is @CPhill.
I do not see any other rationalities why it is posted to this forum.
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