A restaurant offers an "appetizer-plate special" consisting of six selections from its list of appetizers. If there are more than 8000 different possible appetizer-plate specials, what is the least possible number of appetizers?
I don't think anyone on this site can do this problem, not even the experts.
C(16,6) = 8008
Therefore, the least possible number of appetizers is 16
To find the least possible number of appetizers, we need to determine the maximum number of selections that can be made for a given number of appetizers. Let's assume that there are x different appetizers.
To calculate the number of combinations of appetizers, we can use the concept of combinations. The total number of combinations is given by:
nCr = n! / (r! * (n-r)!)
Here, n represents the total number of items, and r represents the number of items selected. In our case, since we are selecting six appetizers out of x different options, it will be xC6.
We know that the number of possible appetizer-plate specials is more than 8000, so:
xC6 > 8000
Now, we need to find the least value of x that satisfies this inequality.
To simplify the calculation, let's make an approximation. We know that 8000 is between 2^12 and 2^13 (4096 and 8192). Therefore, we can start with x = 13 and continue to increase it until we find a value that satisfies the inequality.
Let's calculate xC6 = (x * (x-1) * (x-2) * (x-3) * (x-4) * (x-5)) / (6 * 5 * 4 * 3 * 2 * 1) for x = 13:
13C6 = (13 * 12 * 11 * 10 * 9 * 8) / (6 * 5 * 4 * 3 * 2 * 1) = 1716
With x = 13, the number of appetizer-plate specials is less than 8000, so we need to increase x. Let's try x = 14:
14C6 = (14 * 13 * 12 * 11 * 10 * 9) / (6 * 5 * 4 * 3 * 2 * 1) = 3003
With x = 14, the number of appetizer-plate specials is still less than 8000. Let's try x = 15:
15C6 = (15 * 14 * 13 * 12 * 11 * 10) / (6 * 5 * 4 * 3 * 2 * 1) = 5005
With x = 15, the number of appetizer-plate specials is still less than 8000. Let's try x = 16:
16C6 = (16 * 15 * 14 * 13 * 12 * 11) / (6 * 5 * 4 * 3 * 2 * 1) = 8008
With x = 16, the number of appetizer-plate specials is exactly 8008, which is more than 8000. This is the least possible number of appetizers that satisfies the given condition.
Therefore, the least possible number of appetizers is 16.