A movie company wants to make movies featuring various monsters. They want each movie to have at least 2 different monsters and no two movies to have the exact same set of monsters. What is the minimum number of monsters the company must use in order to make a million movies?
If they have n>=2k monsters to choose from, then we need
C(n,2) + C(n,3) + C(n,4) + ... + C(n,k) >= 500,000
10
∑ C(20,k) = 616,665
k=1
20
To answer this question, we need to find the minimum number of monsters required to create a million movies, ensuring that each movie has at least 2 different monsters and no two movies have the exact same set of monsters.
Let's break down the problem step by step:
1. First, let's consider the number of unique sets of monsters required. Since we want no two movies to have the same set of monsters, each movie must have a unique combination of monsters. So, we need to find the number of unique combinations possible.
2. To calculate the number of unique combinations, we can use the formula for combinations, which is given by: C(n, r) = n! / (r! * (n - r)!), where n is the total number of objects to choose from, and r is the number of objects we want to choose at a time.
3. In our case, n represents the total number of monsters, and r is 2 since we want each movie to have at least 2 different monsters. Therefore, the formula becomes: C(n, 2) = n! / (2! * (n - 2)!).
4. To proceed, let's simplify the formula. We can rewrite 2! as 2 x 1, and (n - 2)! as (n - 2) x (n - 3) x ... x 1. This simplification gives us the following equation: C(n, 2) = n! / (2 x (n - 2)!).
5. Now, we need to find the value of n such that the number of unique combinations equals one million (1,000,000). We can rewrite this equation as follows: n! / (2 x (n - 2)!) = 1,000,000.
6. Let's solve this equation to find the minimum number of monsters required. We can simplify the equation by canceling out the common terms between the numerator and denominator: n x (n - 1) / 2 = 1,000,000.
7. Rearranging the equation, we get: n x (n - 1) = 2 x 1,000,000.
8. Expanding the equation, we have: n^2 - n - 2,000,000 = 0.
9. Now, we can solve this quadratic equation using the quadratic formula: n = (-b ± sqrt(b^2 - 4ac)) / (2a). In this equation, a = 1, b = -1, and c = -2,000,000.
10. Substituting the values into the formula and solving, we find that n = 1,416.31 (approx.).
Therefore, the minimum number of monsters the company must use in order to make a million movies is approximately 1,416.