A bag contains 4 red, 6 white and 9 green marbles. How many ways can 3 red, 2 white and 1 green marbles be selected? Do you used permutation or combination, give your reasoning.
To find the number of ways to select 3 red, 2 white, and 1 green marble from the given bag, we need to determine whether we should use permutations or combinations.
Permutations are used when the order in which the elements are selected matters. Combinations, on the other hand, are used when the order does not matter.
In this case, we are not concerned with the order in which the marbles are selected, but rather with the total number of marbles of each color. Therefore, we should use combinations.
The formula for combinations is given by:
C(n, r) = n! / (r! * (n-r)!),
where n is the total number of items and r is the number of items to be selected.
In our case, we need to calculate:
C(4, 3) * C(6, 2) * C(9, 1).
C(4, 3) represents the number of ways to select 3 red marbles from 4 red marbles. This is calculated as:
4! / (3! * (4-3)!) = 4! / (3! * 1!) = 4.
Similarly, C(6, 2) represents the number of ways to select 2 white marbles from 6 white marbles. This is calculated as:
6! / (2! * (6-2)!) = 6! / (2! * 4!) = 15.
And finally, C(9, 1) represents the number of ways to select 1 green marble from 9 green marbles. This is calculated as:
9! / (1! * (9-1)!) = 9! / (1! * 8!) = 9.
Therefore, the total number of ways to select 3 red, 2 white, and 1 green marble is:
4 * 15 * 9 = 540.
So, there are 540 ways to select the specified combination of marbles from the given bag.