DP9 binary Coefficient Binomial Coefficient @ geeksforgeeks

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 shocould return 6 for n = 4 and k = 2, and it shoshould 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

2) Overlapping Subproblems
Following is simple recursive implementation that simply follows the recursive structure mentioned above.

Package DP; public class BionomialCoefficient {public static void main (String [] args) {int n = 5, k = 2; System. out. println (binomialCoeffRec (n, k); System. out. println (binomialCoeffDP_2D (n, k); System. out. println (binomialCoeffDP_1D (n, k);} public static int binomialCoeffRec (int n, int k) {if (k = 0 | k = n) {return 1;} return binomialCoeffRec (n-1, k-1) + binomialCoeffRec (n-1, k);} // Returns value of Binomial Coefficient C (n, k) // Time Complexity: O (n * k), Auxiliary Space: O (n * k) public static int binomialCoeffDP_2D (int n, int k) {int [] [] dp = new int [n + 1] [k + 1]; // Caculate value of Binomial Coefficient in bottom up mannerfor (int I = 0; I <= n; I ++) {for (int j = 0; j <= Math. min (I, k); j ++) {if (j = 0 | j = I) {// Base Casesdp [I] [j] = 1 ;} else {// Calculate value using previsly stored valuesdp [I] [j] = dp [I-1] [J-1] + dp [I-1] [j] ;}} return dp [n] [k];} // A space optimized Dynamic Programming Solution // Time Complexity: O (n * k), Auxiliary Space: O (k) public static int binomialCoeffDP_1D (int n, int k) {int [] dp = new int [k + 1]; dp [0] = 1; for (int I = 1; I <= n; I ++) {for (int j = Math. min (I, k); j> 0; j --) {// reverse dp [j] + = dp [J-1] ;}} return dp [k] ;}}

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