A research group recruits 10 volunteers to try a new brand of tennis shoes. Only 5 volunteers will be selected to participate in the study.
How many ways can the research group select 5 participants?
10C5 ... 10! / [(10 - 5)! * 5!]
Combinations of n things taken r at a time (without regard to arrangement within group) is:
C(n,r) = n! / [ r! (n-r)! ]
here
10! / [ 5! (5!) ]
= 9*8*7*6 / [ 5*4*3*2]
= 9 * 2 * 7
To determine the number of ways the research group can select 5 participants from 10 volunteers, we can use the combination formula.
The combination formula is given by:
C(n, k) = n! / (k!(n - k)!)
Where:
- n represents the total number of volunteers (10 in this case).
- k represents the number of participants to be selected (5 in this case).
- ! denotes a factorial.
Plug in the values into the formula:
C(10, 5) = 10! / (5!(10 - 5)!)
Simplify:
C(10, 5) = 10! / (5! * 5!)
To further simplify, we can calculate the factorials:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
Now, substitute the factorials into the formula:
C(10, 5) = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (5 * 4 * 3 * 2 * 1))
Perform the calculations:
C(10, 5) = 10 * 9 * 8 * 7 * 6 / (5 * 4 * 3 * 2 * 1)
Cancelling out common terms:
C(10, 5) = 10 * 9 * 8 * 7 * 6 / (5!)
Calculate:
C(10, 5) = 252
Therefore, there are 252 ways the research group can select 5 participants from 10 volunteers.