Questions: To install OTTO (one sale, one must correctly solve 6 numbers from 1 through 65). The order in which the solution is made does not matter. How many different selections of 6 numbers from 1 through 65 are possible?

Solution Steps

To determine the number of different selections of 6 numbers from 1 through 65, we need to calculate the number of combinations. This is a classic combinatorics problem where the order of selection does not matter. We use the combination formula, which is given by C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to choose from, and k is the number of items to choose.

We need to find the number of different selections of 6 numbers from a set of 65 numbers. This is a combinatorial problem where the order of selection does not matter.

The number of combinations of selecting \( k \) items from \( n \) items is given by the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] In our case, \( n = 65 \) and \( k = 6 \).

Substituting the values into the formula, we have: \[ C(65, 6) = \frac{65!}{6!(65-6)!} = \frac{65!}{6! \cdot 59!} \] Calculating this gives us: \[ C(65, 6) = 82598880 \]

Final Answer