There are 5 rotten apples in a crate of 25 apples. How many samples of 3 can
be drawn from the crate?
There are 25!/[3!*22!) = 25*24*23/6 = 23,000 combinations of ways that three apples can be drawn, regardless of how many are rotten.
Are you sure you are asking the right question? If you want the probability that one is rotten in a sample of three, that would be
(5/25)*(22/24)(21/23)*3 = 50.2%
The probability that none of the three drawn (at random) are rotten is 49.6%
Well, if you're looking for rotting apple smoothies, you might want to think twice! But to answer your question, you can use the combination formula. The number of samples of 3 that can be drawn from a crate of 25 apples, including the rotten ones, is determined by 25C3. So hold your nose and do some math!
To calculate the number of samples of 3 that can be drawn from a crate of 25 apples, we will use the concept of combinations.
In this scenario, since there are 5 rotten apples in the crate, we need to consider them as undesirable items and exclude them from the sample. We only want to select 3 apples from the 25 available non-rotten apples in the crate.
The formula to calculate the number of combinations is:
nCr = n! / r!(n - r)!
Where:
n is the total number of items (non-rotten apples in this case)
r is the number of items you want to select (3 in this case)
! denotes the factorial, which means multiplying all the whole numbers from 1 to that number
Plugging in the values, we have:
n = 25
r = 3
nCr = 25! / 3!(25 - 3)!
= 25! / 3!22!
Now, let's calculate the factorial for each:
25! = 25 x 24 x 23 x 22 x 21 x ... x 3 x 2 x 1
3! = 3 x 2 x 1
22! = 22 x 21 x ... x 3 x 2 x 1
Substituting these values back into the formula:
nCr = (25 x 24 x 23 x 22 x 21 x ... x 3 x 2 x 1) / (3 x 2 x 1) x (22 x 21 x ... x 3 x 2 x 1)
Now, we can simplify the expression:
nCr = (25 x 24 x 23) / (3 x 2 x 1)
nCr = (138,000) / (6)
nCr = 23,000
Therefore, there are 23,000 samples of 3 that can be drawn from the crate.
To find the number of samples of 3 that can be drawn from the crate, we can use the concept of combinations. A combination is a selection of items without regard to the order in which they are selected.
The formula for calculating combinations is given by nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items being selected.
In this case, we have 25 apples in the crate, and we want to select 3 apples at a time. So, n = 25 and r = 3.
To calculate the combinations, we can substitute the values into the formula:
25C3 = 25! / (3! * (25-3)!)
Now let's break down the calculation step by step:
1. Calculate 25!
25! = 25 * 24 * 23 * ... * 3 * 2 * 1
2. Calculate 3!
3! = 3 * 2 * 1
3. Calculate (25-3)!
(25-3)! = 22!
4. Substitute the values into the formula:
25C3 = 25! / (3! * (25-3)!)
= (25 * 24 * 23 * ... * 3 * 2 * 1) / ((3 * 2 * 1) * (22 * 21 * ... * 3 * 2 * 1))
Many of the terms in the numerator and denominator will cancel each other out, simplifying the calculation.
Thus, 25C3 = (25 * 24 * 23) / (3 * 2 * 1)
= 2300 / 6
= 383.33
So, there can be 383 samples of 3 drawn from the crate of 25 apples. However, since we cannot have a fraction of a sample, the actual number of samples will be 383.