What is the probability that the sum of two different numbers randomly picked (without replacement) from the set `S = {1,2,3,4}` is 5 ?

Options

A. `1//5`
B. `3//16`
C. `1//4`
D. `1//3`

Answer

To find the probability that the sum of two different numbers randomly picked (without replacement) from the set \( S = \{1, 2, 3, 4\} \) is 5, we can follow these steps:

### Step 1: Determine the total number of outcomes
We need to find the total number of ways to choose 2 different numbers from the set \( S \). The number of combinations of choosing 2 numbers from 4 can be calculated using the combination formula:

\[ \text{Total Outcomes} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 4 \) and \( r = 2 \):

\[ \text{Total Outcomes} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 2: List the favorable outcomes
Next, we need to find the pairs of numbers that sum up to 5. The possible pairs from the set \( S \) are:

- (1, 2)
- (1, 3)
- (1, 4)
- (2, 3)
- (2, 4)
- (3, 4)

Now, we check which of these pairs sum to 5:

- \( 1 + 4 = 5 \)
- \( 2 + 3 = 5 \)

Thus, the favorable outcomes are (1, 4) and (2, 3). Therefore, there are 2 favorable outcomes.

### Step 3: Calculate the probability
The probability \( P \) of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ P(\text{sum} = 5) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer
The probability that the sum of two different numbers randomly picked from the set \( S \) is 5 is \( \frac{1}{3} \).

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