Merge pull request #21 from alberthu16/day27

Day 27: pascal's triangle

Author: Albert Hu

day27/README.md

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+question of the day: https://codefights.com/challenge/szuSfd9RjD3jwKtwj
+
+If we write out all the [binomial coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient)
+`C(N, K)` for all `N ≤ 5`, here's what we'll get:
+
+``` python
+N = 0: [1]
+N = 1: [1, 1]
+N = 2: [1, 2, 1]
+N = 3: [1, 3, 3, 1]
+N = 4: [1, 4, 6, 4, 1]
+N = 5: [1, 5, 10, 10, 5, 1]
+```
+
+This set of lists is often referred to as [Pascal's Triangle](https://en.wikipedia.org/wiki/Pascal%27s_triangle).
+As we can see, 17 out of 21 numbers in it are not divisible by 5.
+
+Given an integer num and a prime number, calculate the number of all the binomial coefficients for all i ≤ num that are not divisible by prime.
+
+Example
+
+For `num = 5` and `prime = 5`, the output should be
+
+`countingBinomialCoefficient(num, prime) = 17`.
+
+## Ideas
+
+Brute force: Calculate Pascal's Triangle, row by row, and keep a counter on how many are
+not divisible by `prime`.
+
+`O(n<sup>2</sup>)` runtime, `O(n)` space, where `n` is the number of binomial coefficients.
+
+## Code
+
+[Python](./countingBinomialCoefficient.py)
+
+## Follow up
+
+Too bad the brute force is too slow for calculations on like 10 million rows.
+Here's a trick: https://cjordan.github.io/2013/12/29/solving-project-euler-148/#fnref:3
+
+FML math is hard

day27/countingBinomialCoefficient.py

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+class BinomialCoefficient(object):
+    def __init__(self):
+        self.coefficients = {}
+        self.notDivisible = {}
+
+    # returns the binomial expansion coefficient for TOTAL choose SUBGROUP
+    # aka C(total, subgroup)
+    def getBinomialCoefficient(self, total, subgroup):
+        if (total, subgroup) in self.coefficients:
+            return self.coefficients[(total, subgroup)]
+
+        if total == subgroup or subgroup == 0:
+            self.coefficients[(total, subgroup)] = 1
+            return 1
+
+        return self.getBinomialCoefficient(total-1, subgroup) + self.getBinomialCoefficient(total-1, subgroup-1)
+
+    # returns the number of coefficients in Pascal's triangle up to
+    # the NUMth row of the triangle that are not divisible by PRIME
+    def binomialCoefficientsNotDivisibleByPrime(self, num, prime):
+        if num == 0:
+            self.notDivisible[prime] = {0: 1}
+            return 1
+
+        if prime in self.notDivisible and num in self.notDivisible[prime]:
+            return self.notDivisible[prime][num]
+
+        notDivisibleThisRow = len([i for i in xrange(num+1) if self.getBinomialCoefficient(num, i) % prime != 0])
+
+        self.notDivisible[prime][num] = self.binomialCoefficientsNotDivisibleByPrime(num-1, prime) + notDivisibleThisRow
+        return self.notDivisible[prime][num]
+
+def testGetBinomialCoefficient(binomialCalculator):
+    b = binomialCalculator
+
+    assert b.getBinomialCoefficient(0, 0) == 1
+
+    assert b.getBinomialCoefficient(1, 0) == 1
+    assert b.getBinomialCoefficient(1, 1) == 1
+
+    assert b.getBinomialCoefficient(2, 0) == 1
+    assert b.getBinomialCoefficient(2, 1) == 2
+    assert b.getBinomialCoefficient(2, 2) == 1
+
+    assert b.getBinomialCoefficient(3, 0) == 1
+    assert b.getBinomialCoefficient(3, 1) == 3
+    assert b.getBinomialCoefficient(3, 2) == 3
+    assert b.getBinomialCoefficient(3, 3) == 1
+
+    assert b.getBinomialCoefficient(4, 0) == 1
+    assert b.getBinomialCoefficient(4, 1) == 4
+    assert b.getBinomialCoefficient(4, 2) == 6
+    assert b.getBinomialCoefficient(4, 3) == 4
+    assert b.getBinomialCoefficient(4, 4) == 1
+
+    assert b.getBinomialCoefficient(5, 0) == 1
+    assert b.getBinomialCoefficient(5, 1) == 5
+    assert b.getBinomialCoefficient(5, 2) == 10
+    assert b.getBinomialCoefficient(5, 3) == 10
+    assert b.getBinomialCoefficient(5, 4) == 5
+    assert b.getBinomialCoefficient(5, 5) == 1
+
+def testCountBinomialCoefficientsNotDivisibleByPrime(binomialCalculator):
+    b = binomialCalculator
+
+    # prime = 2
+    assert b.binomialCoefficientsNotDivisibleByPrime(0, 2) == 1
+    assert b.binomialCoefficientsNotDivisibleByPrime(1, 2) == 3
+    assert b.binomialCoefficientsNotDivisibleByPrime(2, 2) == 5
+    assert b.binomialCoefficientsNotDivisibleByPrime(3, 2) == 9
+    assert b.binomialCoefficientsNotDivisibleByPrime(4, 2) == 11
+    assert b.binomialCoefficientsNotDivisibleByPrime(5, 2) == 15
+
+    # prime = 5
+    assert b.binomialCoefficientsNotDivisibleByPrime(0, 5) == 1
+    assert b.binomialCoefficientsNotDivisibleByPrime(1, 5) == 3
+    assert b.binomialCoefficientsNotDivisibleByPrime(2, 5) == 6
+    assert b.binomialCoefficientsNotDivisibleByPrime(3, 5) == 10
+    assert b.binomialCoefficientsNotDivisibleByPrime(4, 5) == 15
+    assert b.binomialCoefficientsNotDivisibleByPrime(5, 5) == 17
+    assert b.binomialCoefficientsNotDivisibleByPrime(6, 5) == 21
+
+    # prime = 7
+    assert b.binomialCoefficientsNotDivisibleByPrime(5, 7) == 21 
+
+    # BIG DATA
+    # assert b.binomialCoefficientsNotDivisibleByPrime(999999999, 7)  == 2129970655314432
+    # assert b.binomialCoefficientsNotDivisibleByPrime(879799878, 17) == 6026990181372288
+    # assert b.binomialCoefficientsNotDivisibleByPrime(879799878, 19) == 8480245105257600
+
+def main():
+    b = BinomialCoefficient()
+    testGetBinomialCoefficient(b)
+    testCountBinomialCoefficientsNotDivisibleByPrime(b)
+
+if __name__ == "__main__":
+    main()