Geeks 面试题：Binomial Coefficient

Binomial Coefficient

Following are common definition of Binomial Coefficients.
1) A binomial coefficient C(n, k) can be defined as the coefficient of X^k in the expansion of (1 + X)^n.

2) A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set.

The Problem
Write a function that takes two parameters n and k and returns the value of Binomial Coefficient C(n, k).For example, your function should return 6 for n = 4 and k = 2, and it should return 10 for n = 5 and k = 2.

1) Optimal Substructure
The value of C(n, k) can recursively calculated using following standard formula for Binomial Cofficients.

   C(n, k) = C(n-1, k-1) + C(n-1, k)   C(n, 0) = C(n, n) = 1

http://www.geeksforgeeks.org/dynamic-programming-set-9-binomial-coefficient/

要清楚计算方法，然后一步一步计算，注意画好表。

下面是动态规划法和递归法。

其中第二个程序是最省内存的方法了。

比如计算C(n,k) = C(10,8)其实C(10,8) = C(10,2)那么就可以利用这个特性，进一步优化空间有8个空间变成2个空间，而且循环也有10*8成了10*2，最省内存和最优化的方法了。

// A space optimized Dynamic Programming Solutionint binomialCoeffDPSaveSpace(int n, int k){int* C = (int*)calloc(k+1, sizeof(int));int i, j, res;C[0] = 1;for(i = 1; i <= n; i++){for(j = min(i, k); j > 0; j--)C[j] = C[j] + C[j-1];}res = C[k];  // Store the result before freeing memoryfree(C);  // free dynamically allocated memory to avoid memory leakreturn res;}int binomialCoDPSaveSpace(int n, int k){k = min(k,n-k);vector<int> ta(k+1);ta[0] = 1;for (int i = 0; i <= n; i++){for (int j = min(i,k); j > 0 ; j--){ta[j] = ta[j] +ta[j-1];}}return ta[k];}// Returns value of Binomial Coefficient C(n, k)int binomialCoeff(int n, int k){// Base Casesif (k==0 || k==n)return 1;// Recurreturn  binomialCoeff(n-1, k-1) + binomialCoeff(n-1, k);}int main(){int n = 10, k = 8;printf("Value of C(%d, %d) is %d\n ", n, k, binomialCoeff(n, k));cout<<binomialCoDPSaveSpace(n, k)<<endl;system("pause");return 0;}