There are 5 pairs of shoes in a cupboard from which 4 shoes are picked at random. The probability that there is at least one pair is

Options

A. `8/21`
B. `11/21`
C. `13/21`
D. `12/31`

Answer

To solve the problem of finding the probability that there is at least one pair of shoes when picking 4 shoes from 5 pairs, we can follow these steps:

### Step 1: Determine the Total Number of Shoes
There are 5 pairs of shoes, which means there are a total of 10 individual shoes (5 left shoes and 5 right shoes).

### Step 2: Calculate the Total Outcomes for Picking 4 Shoes
The total number of ways to choose 4 shoes from 10 shoes can be calculated using the combination formula:
\[ \text{Total Outcomes} = \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

### Step 3: Calculate the Favorable Outcomes for Picking Shoes Without Any Pairs
To find the probability of picking shoes without any pairs, we need to select 4 shoes such that no two shoes are from the same pair.

1. **Choose 4 pairs from the 5 available pairs**:
\[ \text{Ways to choose 4 pairs} = \binom{5}{4} = 5 \]

2. **For each chosen pair, select one shoe (either left or right)**:
Since we have 4 pairs chosen, and for each pair, we can choose either the left shoe or the right shoe, we have:
\[ \text{Ways to choose one shoe from each of the 4 pairs} = 2^4 = 16 \]

3. **Total ways to pick 4 shoes without any pairs**:
\[ \text{Favorable Outcomes} = \text{Ways to choose 4 pairs} \times \text{Ways to choose one shoe from each pair} = 5 \times 16 = 80 \]

### Step 4: Calculate the Probability of Picking at Least One Pair
The probability of picking at least one pair is the complement of the probability of picking no pairs. We can calculate it as follows:
\[ \text{Probability of no pairs} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{80}{210} \]
\[ \text{Probability of at least one pair} = 1 - \text{Probability of no pairs} = 1 - \frac{80}{210} = \frac{210 - 80}{210} = \frac{130}{210} \]
Now, we can simplify this fraction:
\[ \frac{130}{210} = \frac{13}{21} \]

### Final Answer
Thus, the probability that there is at least one pair when picking 4 shoes is:
\[ \frac{13}{21} \]