in how many ways can 4 different prizes be given to any 4 of 10 people if no person receives more than 1 prize?
help please and explain!!!!!!
To solve this problem, we can use the concept of combinations.
First, let's understand what is being asked. We have 4 different prizes and we need to distribute them among 10 people in such a way that no person receives more than 1 prize.
Since each person can receive at most 1 prize, we need to choose 4 people out of the 10 to give them the prizes. The order in which the prizes are given doesn't matter, as long as each person gets only one.
To calculate the number of ways to choose 4 people out of 10, we can use the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where:
- n is the total number of objects (in this case, 10 people).
- r is the number of objects chosen (in this case, 4 people).
- ! denotes the factorial, which is the product of all positive integers less than or equal to the given number.
Using this formula, we can calculate the number of ways as follows:
C(10, 4) = 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
= (10 * 9 * 8 * 7 * 6!) / (4! * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
Therefore, there are 210 different ways to give 4 different prizes to any 4 of the 10 people, considering that no person receives more than 1 prize.