Combinations: If you are selecting 5 persons out of 10 to compete in a hot dog eating contest, there are ______ different ways that people can be organised. The order that they are chosen doesn't matter this is a combination.
Please help.
consider 10C5
10C5 = 10! / [(10 - 5)! * 5!]
Thank you.
To find the number of different ways to select 5 persons out of 10, we can use the concept of combinations.
The formula for combinations is given by:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of options to choose from, and r is the number of selections we want to make.
In this case, n = 10 (total number of persons) and r = 5 (persons to be selected).
Plugging in the values, we get:
C(10, 5) = 10! / (5! * (10-5)!)
Calculating this expression step by step:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
(10 - 5)! = 5! = 5 * 4 * 3 * 2 * 1
Now, substitute these values:
C(10, 5) = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / [(5 * 4 * 3 * 2 * 1) * (5 * 4 * 3 * 2 * 1)]
Simplifying further:
C(10, 5) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)
Cancelling out common factors:
C(10, 5) = (10 * 9 * 8 * 7 * 6) / (5!)
Calculating the numerator:
(10 * 9 * 8 * 7 * 6) = 30,240
Calculating the denominator:
5! = 5 * 4 * 3 * 2 * 1 = 120
Finally:
C(10, 5) = 30,240 / 120 = 252
Therefore, there are 252 different ways that people can be organized for the hot dog eating contest.