Final answer:
Archie can choose from 16 different groups of 6 entrees from the 16 available, using the combination formula C(16, 6). This is calculated by dividing 16 factorial by the product of 6 factorial and 10 factorial.
Step-by-step explanation:
To find out how many groups of 6 entrees Archie can choose from the 16 available at the restaurant, where the order of entrees chosen does not matter, we use the combination formula. This is a problem of combinatorics which falls under the mathematics subject. The formula for combinations is C(n, k) = n! / (k! * (n-k)!), where 'n' represents the total number of items to choose from, in this case, 16 entrees, and 'k' represents the number of items to choose, which is 6 entrees.
Let's calculate the number of combinations:
C(16, 6) = 16! / (6! * (16-6)!) = 16! / (6! * 10!) = 8008 / 720 = 11,440 /720 = 16
The 16 factorial (16!) represents the total number of ways to arrange 16 items, the 6 factorial (6!) represents the number of ways to arrange 6 items within a group, and the 10 factorial (10!) represents the number of ways to arrange the remaining 10 items. When we calculate this using the combination formula, we get 16 combinations. Therefore, Archie can choose from 16 different groups of 6 entrees.