Suppose you have a lock that consists of 5 digits that are randomly chosen from the set {0,1,2,3,4,5,6,7,8,9}. Write a C++ program to demonstrate how to find the correct lock combination by using recursion and backtracking. Show step-by-step how to solve the problem. Draw three recursive level calls example for the recursion tree call with input values. Note that, the initial state of the key is {0,0,0,0,0} and the lock combination is randomly chosen {rand(0-9), rand(0-9), rand(0-9), rand(0-9), ), rand(0-9)}.

Sure, I can help you with that! Here's a C++ program that uses recursion and backtracking to find the correct lock combination:

#include <iostream>
#include <vector>
#include <cstdlib>
#include <ctime>

using namespace std;

const int NUM_DIGITS = 5;

void generateRandomCombination(vector<int>& combination) {
    for (int i = 0; i < NUM_DIGITS; i++) {
        combination.push_back(rand() % 10);
    }
}

void printCombination(const vector<int>& combination) {
    for (int i = 0; i < NUM_DIGITS; i++) {
        cout << combination[i];
    }
    cout << endl;
}

bool tryCombination(vector<int>& currentCombination, const vector<int>& targetCombination, int digitIndex) {
    if (digitIndex == NUM_DIGITS) {
        return currentCombination == targetCombination;
    }
    for (int i = 0; i < 10; i++) {
        currentCombination[digitIndex] = i;
        if (tryCombination(currentCombination, targetCombination, digitIndex + 1)) {
            return true;
        }
    }
    return false;
}

int main() {
    srand(time(0));
    vector<int> targetCombination;
    generateRandomCombination(targetCombination);
    cout << "Target combination: ";
    printCombination(targetCombination);
    vector<int> currentCombination(NUM_DIGITS, 0);
    if (tryCombination(currentCombination, targetCombination, 0)) {
        cout << "Found the correct combination: ";
        printCombination(currentCombination);
    } else {
        cout << "Unable to find the correct combination." << endl;
    }
    return 0;
}

The `generateRandomCombination` function generates a random lock combination and stores it in a vector. The `printCombination` function simply prints out a given combination.

The `tryCombination` function takes in the current combination being tested, the target combination to be found, and the current digit index that is being tested. It recursively tries all possible combinations of digits for the current index, and returns true if the correct combination is found.

In the `main` function, we first generate a random target combination, print it out, and initialize the current combination to all zeros. We then call `tryCombination` with the current combination, target combination, and starting index of 0. If the correct combination is found, we print it out. Otherwise, we print a message saying that we were unable to find the correct combination.

Here are three example recursive level calls for the recursion tree call with input values:

tryCombination({0, 0, 0, 0, 0}, {3, 2, 5, 8, 1}, 0)
├──tryCombination({0, 0, 0, 0, 3}, {3, 2, 5, 8, 1}, 1)
│  ├──tryCombination({0, 0, 0, 2, 3}, {3, 2, 5, 8, 1}, 2)
│  │  ├──tryCombination({0, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│  │  │  ├──tryCombination({3, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│  │  │  └──tryCombination({4, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
│  │  └──tryCombination({0, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│  │     ├──tryCombination({3, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│  │     └──tryCombination({4, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
│  └──tryCombination({0, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│     ├──tryCombination({3, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│     └──tryCombination({4, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
├──tryCombination({0, 0, 0, 0, 4}, {3, 2, 5, 8, 1}, 1)
│  ├──tryCombination({0, 0, 0, 2, 4}, {3, 2, 5, 8, 1}, 2)
│  │  ├──tryCombination({0, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│  │  │  ├──tryCombination({3, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│  │  │  └──tryCombination({4, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
│  │  └──tryCombination({0, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│  │     ├──tryCombination({3, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│  │     └──tryCombination({4, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
│  └──tryCombination({0, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│     ├──tryCombination({3, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│     └──tryCombination({4, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
└──tryCombination({0, 0, 0, 0, 5}, {3, 2, 5, 8, 1}, 1)
   ├──tryCombination({0, 0, 0, 2, 5}, {3, 2, 5, 8, 1}, 2)
   │  ├──tryCombination({0, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
   │  │  ├──tryCombination({3, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
   │  │  └──tryCombination({4, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)
   │  └──tryCombination({0, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
   │     ├──tryCombination({3, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
   │     └──tryCombination({4, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)
   └──tryCombination({0, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
      ├──tryCombination({3, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
      └──tryCombination({4, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)

In this example, we're trying to find the correct combination for the lock with the target combination {3, 2, 5, 8, 1}. The recursion tree shows the different combinations being tried at each recursive call. At each level, the function tries all possible digits for the current index, and recursively calls itself with the next digit index. If a combination is found that matches the target combination, the function returns true and stops recursing.