A, B and C were given a problem in Mathematics whose respective probabilities of solving it are `(1)/(2),(1)/(3) and (1)/(4)`. The probabiltiy that A alone solves it is ______

Options

A. `(1)/(4)`
B. `(3)/(4)`
C. `(11)/(24)`
D. `(17)/(24)`

Answer

To find the probability that A alone solves the problem while B and C do not solve it, we can follow these steps:

### Step 1: Identify the probabilities
- Probability that A solves the problem, \( P(A) = \frac{1}{2} \)
- Probability that B solves the problem, \( P(B) = \frac{1}{3} \)
- Probability that C solves the problem, \( P(C) = \frac{1}{4} \)

### Step 2: Calculate the probabilities that B and C do not solve the problem
- The probability that B does not solve the problem, \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- The probability that C does not solve the problem, \( P(C') = 1 - P(C) = 1 - \frac{1}{4} = \frac{3}{4} \)

### Step 3: Use the independence of events
Since A, B, and C are independent in solving the problem, the probability that A solves it while B and C do not can be calculated as:
\[ P(A \text{ and } B' \text{ and } C') = P(A) \times P(B') \times P(C') \]

### Step 4: Substitute the values
Substituting the values we calculated:
\[ P(A \text{ and } B' \text{ and } C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \]

### Step 5: Calculate the final probability
Now, we can simplify the expression:
\[ P(A \text{ and } B' \text{ and } C') = \frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{1 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 4} = \frac{6}{24} = \frac{1}{4} \]

Thus, the probability that A alone solves the problem is:
\[ \boxed{\frac{1}{4}} \]