Ten pairs of shoes are in a closet. Four shoes are selected at random. Find the probability that there is at least one pair among the four selected.

Answer

To solve the problem of finding the probability that there is at least one pair among the four selected shoes from ten pairs, we can follow these steps:

### Step 1: Understand the Total Number of Shoes
We have 10 pairs of shoes, which means there are a total of \( 10 \times 2 = 20 \) shoes.

### Step 2: Calculate the Total Ways to Select 4 Shoes
The total number of ways to select 4 shoes from 20 is given by the combination formula:
\[ \text{Total ways} = \binom{20}{4} \]
Calculating this:
\[ \binom{20}{4} = \frac{20!}{4!(20-4)!} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845 \]

### Step 3: Calculate the Number of Ways to Select 4 Shoes with No Pairs
To find the number of ways to select 4 shoes such that there are no pairs, we first select 4 different pairs from the 10 pairs, and then select 1 shoe from each of the selected pairs.

1. Choose 4 pairs from 10:
\[ \text{Ways to choose pairs} = \binom{10}{4} \]
Calculating this:
\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

2. For each selected pair, there are 2 choices (left shoe or right shoe). Therefore, for 4 pairs:
\[ \text{Ways to choose shoes} = 2^4 = 16 \]

3. Thus, the total number of ways to select 4 shoes with no pairs is:
\[ \text{Total ways with no pairs} = \binom{10}{4} \times 2^4 = 210 \times 16 = 3360 \]

### Step 4: Calculate the Probability of Selecting at Least One Pair
The probability of selecting at least one pair is given by:
\[ P(\text{at least one pair}) = 1 - P(\text{no pairs}) \]
Where \( P(\text{no pairs}) \) is the ratio of the number of ways to select 4 shoes with no pairs to the total ways to select 4 shoes:
\[ P(\text{no pairs}) = \frac{\text{Ways with no pairs}}{\text{Total ways}} = \frac{3360}{4845} \]

Now, substituting this into the probability of at least one pair:
\[ P(\text{at least one pair}) = 1 - \frac{3360}{4845} = \frac{4845 - 3360}{4845} = \frac{1485}{4845} \]

### Step 5: Simplify the Probability
To simplify \( \frac{1485}{4845} \):
\[ \text{GCD of 1485 and 4845} = 15 \]
Dividing both numerator and denominator by 15:
\[ \frac{1485 \div 15}{4845 \div 15} = \frac{99}{323} \]

### Final Answer
Thus, the probability that there is at least one pair among the four selected shoes is:
\[ \boxed{\frac{99}{323}} \]