Jeff has a box containing 26 tiles, each with a different letter of the alphabet. He randomly draws 4 letters, one at a time without replacement.
What is the probability that he will choose the letters N, W, B, T?
A. (26-4)!/26!
B. 4!/26!
C. 4!(26-4)!/26!
D. 1/26!
1/26 * 1/25 * 1/24 * 1/23 = 22!/26!
To find the probability of Jeff choosing the letters N, W, B, and T in that specific order, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
The total number of possible outcomes is the number of ways Jeff can choose 4 letters out of the 26 available. This can be calculated using the combination formula, which is denoted as nCr. In this case, n (the total number of letters) is 26 and r (the number of letters Jeff is choosing) is 4.
The formula for combination is: nCr = n! / (r! * (n-r)!)
Plugging in the values:
26C4 = 26! / (4! * (26-4)!)
Now, we need to calculate the number of favorable outcomes, which is just 1 since we want Jeff to pick the specific letters N, W, B, and T in that order.
Therefore, the probability can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 26C4
Simplifying the expression:
Probability = 1 / (26! / (4! * (26-4)!))
Hence, the correct answer is option D) 1 / 26!.
To find the probability of Jeff choosing the letters N, W, B, T in that specific order, we need to evaluate the number of favorable outcomes (arrangements of those 4 letters) divided by the total number of possible outcomes.
1. The number of favorable outcomes:
Since Jeff is drawing the letters one at a time without replacement, the probability of each letter being drawn is dependent on the previous draws. Therefore, we need to consider the ordering of the letters.
The first letter Jeff chooses has a probability of 1, as any letter can be chosen initially. After that, the probability of drawing the second letter is 1 out of 25 remaining letters. Similarly, the probability of drawing the third letter is 1 out of 24 remaining letters, and the probability of drawing the fourth letter is 1 out of 23 remaining letters.
So, the number of favorable outcomes is 1 * 1/25 * 1/24 * 1/23.
2. The total number of possible outcomes:
Jeff is drawing 4 letters from a box containing 26 tiles, with no replacement. So, the total number of possible outcomes is the number of ways to arrange 4 letters out of the 26 available, which can be calculated using the formula for combinations:
26 choose 4 = 26! / (4! * (26-4)!)
Now, let's simplify the expressions:
1. The number of favorable outcomes: 1 * 1/25 * 1/24 * 1/23 = 1/(25 * 24 * 23)
2. The total number of possible outcomes: 26! / (4! * (26-4)!) = 26! / (4! * 22!)
Finally, we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = (1/(25 * 24 * 23)) / (26! / (4! * 22!))
= (1/(25 * 24 * 23)) * ((4! * 22!) / 26!)
So, the correct answer would be option D. 1/26!.