Added solution for euler problem 493 (#5573)

* Added solution for problem 493

* fixed typo

* return result as string

Author: skief

Diff for: project_euler/problem_493/__init__.py

@@ -0,0 +1,53 @@
+"""
+Project Euler Problem 493: https://projecteuler.net/problem=493
+
+70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.
+What is the expected number of distinct colours in 20 randomly picked balls?
+Give your answer with nine digits after the decimal point (a.bcdefghij).
+
+-----
+
+This combinatorial problem can be solved by decomposing the problem into the
+following steps:
+1. Calculate the total number of possible picking cominations
+[combinations := binom_coeff(70, 20)]
+2. Calculate the number of combinations with one colour missing
+[missing := binom_coeff(60, 20)]
+3. Calculate the probability of one colour missing
+[missing_prob := missing / combinations]
+4. Calculate the probability of no colour missing
+[no_missing_prob := 1 - missing_prob]
+5. Calculate the expected number of distinct colours
+[expected = 7 * no_missing_prob]
+
+References:
+- https://en.wikipedia.org/wiki/Binomial_coefficient
+"""
+
+import math
+
+BALLS_PER_COLOUR = 10
+NUM_COLOURS = 7
+NUM_BALLS = BALLS_PER_COLOUR * NUM_COLOURS
+
+
+def solution(num_picks: int = 20) -> str:
+    """
+    Calculates the expected number of distinct colours
+
+    >>> solution(10)
+    '5.669644129'
+
+    >>> solution(30)
+    '6.985042712'
+    """
+    total = math.comb(NUM_BALLS, num_picks)
+    missing_colour = math.comb(NUM_BALLS - BALLS_PER_COLOUR, num_picks)
+
+    result = NUM_COLOURS * (1 - missing_colour / total)
+
+    return f"{result:.9f}"
+
+
+if __name__ == "__main__":
+    print(solution(20))