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Eggs dropping puzzle (Binomial Coefficient and Binary Search Solution)
• Difficulty Level : Expert
• Last Updated : 29 May, 2021

Given n eggs and k floors, find the minimum number of trials needed in worst case to find the floor below which all floors are safe. A floor is safe if dropping an egg from it does not break the egg. Please see n eggs and k floors. for complete statements

Example

```Input : n = 2, k = 10
Output : 4
We first try from 4-th floor. Two cases arise,
(1) If egg breaks, we have one egg left so we
need three more trials.
(2) If egg does not break, we try next from 7-th
floor. Again two cases arise.
We can notice that if we choose 4th floor as first
floor, 7-th as next floor and 9 as next of next floor,
we never exceed more than 4 trials.

Input : n = 2. k = 100
Output : 14```

We have discussed the problem for 2 eggs and k floors. We have also discussed a dynamic programming solution to find the solution. The dynamic programming solution is based on below recursive nature of the problem. Let us look at the discussed recursive formula from a different perspective.
How many floors we can cover with x trials?
When we drop an egg, two cases arise.

1. If egg breaks, then we are left with x-1 trials and n-1 eggs.
2. If egg does not break, then we are left with x-1 trials and n eggs
```Let maxFloors(x, n) be the maximum number of floors
that we can cover with x trials and n eggs. From above
two cases, we can write.

maxFloors(x, n) = maxFloors(x-1, n-1) + maxFloors(x-1, n) + 1
For all x >= 1 and n >= 1

Base cases :
We can't cover any floor with 0 trials or 0 eggs
maxFloors(0, n) = 0
maxFloors(x, 0) = 0

Since we need to cover k floors,
maxFloors(x, n) >= k           ----------(1)

The above recurrence simplifies to following,
Refer this for proof.

maxFloors(x, n) = &Sum;xCi
1 <= i <= n   ----------(2)
Here C represents Binomial Coefficient.

From above two equations, we can say.
&Sum;xCj  >= k
1 <= i <= n
Basically we need to find minimum value of x
that satisfies above inequality. We can find
such x using Binary Search.```

## C++

 `// C++ program to find minimum``// number of trials in worst case.``#include ` `using` `namespace` `std;` `// Find sum of binomial coefficients xCi``// (where i varies from 1 to n).``int` `binomialCoeff(``int` `x, ``int` `n, ``int` `k)``{``    ``int` `sum = 0, term = 1;``    ``for` `(``int` `i = 1; i <= n; ++i) {``        ``term *= x - i + 1;``        ``term /= i;``        ``sum += term;``          ``if``(sum>k)``          ``return` `sum;``    ``}``    ``return` `sum;``}` `// Do binary search to find minimum``// number of trials in worst case.``int` `minTrials(``int` `n, ``int` `k)``{``    ``// Initialize low and high as 1st``    ``// and last floors``    ``int` `low = 1, high = k;` `    ``// Do binary search, for every mid,``    ``// find sum of binomial coefficients and``    ``// check if the sum is greater than k or not.``    ``while` `(low < high) {``        ``int` `mid = (low + high) / 2;``        ``if` `(binomialCoeff(mid, n, k) < k)``            ``low = mid + 1;``        ``else``            ``high = mid;``    ``}` `    ``return` `low;``}` `/* Driver code*/``int` `main()``{``    ``cout << minTrials(2, 10);``    ``return` `0;``}`

## Java

 `// Java program to find minimum``// number of trials in worst case.``class` `Geeks {` `// Find sum of binomial coefficients xCi``// (where i varies from 1 to n). If the sum``// becomes more than K``static` `int` `binomialCoeff(``int` `x, ``int` `n, ``int` `k)``{``    ``int` `sum = ``0``, term = ``1``;``    ``for` `(``int` `i = ``1``; i <= n && sum < k; ++i) {``        ``term *= x - i + ``1``;``        ``term /= i;``        ``sum += term;``    ``}``    ``return` `sum;``}` `// Do binary search to find minimum``// number of trials in worst case.``static` `int` `minTrials(``int` `n, ``int` `k)``{``    ``// Initialize low and high as 1st``    ``//and last floors``    ``int` `low = ``1``, high = k;` `    ``// Do binary search, for every mid,``    ``// find sum of binomial coefficients and``    ``// check if the sum is greater than k or not.``    ``while` `(low < high) {``        ``int` `mid = (low + high) / ``2``;``        ``if` `(binomialCoeff(mid, n, k) < k)``            ``low = mid + ``1``;``        ``else``            ``high = mid;``    ``}` `    ``return` `low;``}` `/* Driver code*/``public` `static` `void` `main(String args[])``{``    ``System.out.println(minTrials(``2``, ``10``));``}``}` `// This code is contributed by ankita_saini`

## Python3

 `# Python3 program to find minimum``# number of trials in worst case.` `# Find sum of binomial coefficients``# xCi (where i varies from 1 to n).``# If the sum becomes more than K``def` `binomialCoeff(x, n, k):` `    ``sum` `=` `0``;``    ``term ``=` `1``;``    ``i ``=` `1``;``    ``while``(i <``=` `n ``and` `sum` `< k):``        ``term ``*``=` `x ``-` `i ``+` `1``;``        ``term ``/``=` `i;``        ``sum` `+``=` `term;``        ``i ``+``=` `1``;``    ``return` `sum``;` `# Do binary search to find minimum``# number of trials in worst case.``def` `minTrials(n, k):` `    ``# Initialize low and high as``    ``# 1st and last floors``    ``low ``=` `1``;``    ``high ``=` `k;` `    ``# Do binary search, for every``    ``# mid, find sum of binomial``    ``# coefficients and check if``    ``# the sum is greater than k or not.``    ``while` `(low < high):` `        ``mid ``=` `int``((low ``+` `high) ``/` `2``);``        ``if` `(binomialCoeff(mid, n, k) < k):``            ``low ``=` `mid ``+` `1``;``        ``else``:``            ``high ``=` `mid;` `    ``return` `int``(low);` `# Driver Code``print``(minTrials(``2``, ``10``));` `# This code is contributed``# by mits`

## C#

 `// C# program to find minimum``// number of trials in worst case.``using` `System;` `class` `GFG``{` `// Find sum of binomial coefficients``// xCi (where i varies from 1 to n).``// If the sum becomes more than K``static` `int` `binomialCoeff(``int` `x,``                         ``int` `n, ``int` `k)``{``    ``int` `sum = 0, term = 1;``    ``for` `(``int` `i = 1;``             ``i <= n && sum < k; ++i)``    ``{``        ``term *= x - i + 1;``        ``term /= i;``        ``sum += term;``    ``}``    ``return` `sum;``}` `// Do binary search to find minimum``// number of trials in worst case.``static` `int` `minTrials(``int` `n, ``int` `k)``{``    ``// Initialize low and high``    ``// as 1st and last floors``    ``int` `low = 1, high = k;` `    ``// Do binary search, for every``    ``// mid, find sum of binomial``    ``// coefficients and check if the``    ``// sum is greater than k or not.``    ``while` `(low < high)``    ``{``        ``int` `mid = (low + high) / 2;``        ``if` `(binomialCoeff(mid, n, k) < k)``            ``low = mid + 1;``        ``else``            ``high = mid;``    ``}` `    ``return` `low;``}` `// Driver Code``public` `static` `void` `Main()``{``    ``Console.WriteLine(minTrials(2, 10));``}``}` `// This code is contributed``// by Akanksha Rai(Abby_akku)`

## PHP

 ``
Output
`4`

Time Complexity : O(n Log k)

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