Solve the problem
In a certain lottery, 6 different numbers between 1 and 13 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is unimportant
This problem involves selecting 6 numbers out of a set of 13 numbers. Since the order of the numbers does not matter, we can use the concept of combinations to solve this problem. The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
Where C(n, r) represents the number of combinations of selecting r items from a set of n items.
In this case, n = 13 and r = 6. Plugging in these values into the formula, we get:
C(13, 6) = 13! / (6!(13-6)!)
= 13! / (6!7!)
Now, let's simplify the calculation.
First, calculate the factorial of 13:
13! = 13 × 12 × 11 × 10 × 9 × 8 × 7!
Next, calculate the factorial of 6:
6! = 6 × 5 × 4 × 3 × 2 × 1
Now, substitute these values back into the equation:
C(13, 6) = (13 × 12 × 11 × 10 × 9 × 8 × 7!) / (6 × 5 × 4 × 3 × 2 × 1 × 7!)
= (13 × 12 × 11 × 10 × 9 × 8) / (6 × 5 × 4 × 3 × 2 × 1)
Finally, cancel out common factors and simplify:
C(13, 6) = (13 × 12 × 11 × 10 × 9 × 8) / (6 × 5 × 4 × 3 × 2 × 1)
= (13 × 12 × 11 × 10 × 9 × 8) / (6!)
= (13 × 12 × 11 × 2 × 5 × 9 × 8) / (6 × 5 × 4 × 3 × 2 × 1)
= (13 × 11 × 2 × 3 × 8)
= 13 × 11 × 16
= 2288
Therefore, there are 2288 different selections possible in this lottery.