Answer :
Let's tackle this problem step-by-step, focusing on finding the probability of picking the letters A, D, and C in that specific order from a set of given letters.
First, note that the letters present in the bag are A, B, C, and D. The question requires us to pick 3 letters out of these 4, in a specific order: A, D, and C.
### Step 1: Determine the Total Number of Permutations
To determine the total number of ways in which we can pick 3 letters out of 4 in any order, we use permutations [tex]\( P(n, k) \)[/tex], where [tex]\( n \)[/tex] is the total number of items to choose from (4 letters), and [tex]\( k \)[/tex] is the number of items to pick (3 letters).
[tex]\[ P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 4 \times 3 \times 2 = 24 \][/tex]
This means there are 24 possible ways to arrange 4 letters taking 3 at a time.
### Step 2: Determine the Number of Favorable Outcomes
The question asks us to find the probability of picking the letters in a specific order: A, D, and C. For a specific order of three letters, there is only 1 favorable outcome (ADC).
### Step 3: Calculate the Probability
The probability [tex]\( P(E) \)[/tex] of an event [tex]\( E \)[/tex] is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of permutations}} \][/tex]
Substituting the values,
[tex]\[ P(\text{Picking ADC}) = \frac{1}{24} \][/tex]
### Final Answer
Thus, the probability that the letters picked in order are A, D, and C is:
[tex]\[ \boxed{\frac{1}{24}} \][/tex]
This corresponds to option A.
First, note that the letters present in the bag are A, B, C, and D. The question requires us to pick 3 letters out of these 4, in a specific order: A, D, and C.
### Step 1: Determine the Total Number of Permutations
To determine the total number of ways in which we can pick 3 letters out of 4 in any order, we use permutations [tex]\( P(n, k) \)[/tex], where [tex]\( n \)[/tex] is the total number of items to choose from (4 letters), and [tex]\( k \)[/tex] is the number of items to pick (3 letters).
[tex]\[ P(4, 3) = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 4 \times 3 \times 2 = 24 \][/tex]
This means there are 24 possible ways to arrange 4 letters taking 3 at a time.
### Step 2: Determine the Number of Favorable Outcomes
The question asks us to find the probability of picking the letters in a specific order: A, D, and C. For a specific order of three letters, there is only 1 favorable outcome (ADC).
### Step 3: Calculate the Probability
The probability [tex]\( P(E) \)[/tex] of an event [tex]\( E \)[/tex] is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of permutations}} \][/tex]
Substituting the values,
[tex]\[ P(\text{Picking ADC}) = \frac{1}{24} \][/tex]
### Final Answer
Thus, the probability that the letters picked in order are A, D, and C is:
[tex]\[ \boxed{\frac{1}{24}} \][/tex]
This corresponds to option A.
