To win at LOTTO in a certain state, one must correctly select 6 numbers from a collection of 52 numbers ( 1 through 52). The order in which the selections is made does not matter. How many different selections are possible? O 720 O 1,440 20,358,520 o 312 18,009,460

To determine the number of different selections possible, we use the combination formula since the order of selection does not matter. The number of ways to choose 6 numbers out of 52 is given by \[ \binom{52}{6} = \frac{52!}{6!(52-6)!} = \frac{52!}{6! \times 46!}. \] We can simplify this expression by writing the numerator as the product of the 6 highest factors and canceling out the common factors in the denominator: \[ \binom{52}{6} = \frac{52 \times 51 \times 50 \times 49 \times 48 \times 47}{6 \times 5 \times 4 \times 3 \times 2 \times 1}. \] Now, calculate the numerator step by step: 1. \(52 \times 51 = 2652\). 2. \(2652 \times 50 = 132600\). 3. \(132600 \times 49 = 6497400\). 4. \(6497400 \times 48 = 311875200\). 5. \(311875200 \times 47 = 14658134400\). Next, calculate the denominator: \[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720. \] Now, divide the numerator by the denominator: \[ \frac{14658134400}{720} = 20358520. \] Thus, the number of different selections possible is \[ \boxed{20358520}. \]