Using the combination formula, the number of ways 17 people could be divided into five groups containing respectively, 2, 6, 5, 3, and 1 people is 345,888,960.
The number of ways to choose 2 people from 17 is 17 choose 2, denoted as C(17, 2) or 17C2. Similarly, we can calculate the number of ways to choose 6 from the remaining 15, 5 from the remaining 9, 3 from the remaining 4, and 1 from the remaining 1.
The formula for combinations is C(n, r) = n! / (r!(n-r)!), where:
The total number of items = n
The number of items to choose = r
Using the above formula, we can calculate the number of ways for each group and then multiply them together to find the total number of ways to divide the 17 people into the specified groups.
The number of ways to divide 17 people into groups of 2, 6, 5, 3, and 1 is given by: C(17, 2) * C(15, 6) * C(9, 5) * C(4, 3) * C(1, 1)
Performing the calculations:
C(17, 2) = 17! / (2!(17-2)!)
= 136
C(15, 6) = 15! / (6!(15-6)!)
= 50,895
C(9, 5) = 9! / (5!(9-5)!)
= 126
C(4, 3) = 4! / (3!(4-3)!)
= 4
C(1, 1) = 1! / (1!(1-1)!)
= 1
Multiplying these values together: 136 * 50,895 * 126 * 4 * 1 = 345,888,960
Thus, we can conclude that there are 345,888,960 ways to divide 17 people into groups containing 2, 6, 5, 3, and 1 people, respectively.