If two coins are tossed five times, thenthe probability of getting 5 heads and 5 tails is

Options

A. `63//256`
B. `1//1024`
C. `2//205`
D. `9//64`

Explanation

To solve the problem of finding the probability of getting exactly 5 heads and exactly 5 tails when two coins are tossed five times, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We are tossing 2 coins 5 times, which is equivalent to tossing 1 coin 10 times. This is because each toss of 2 coins can result in either heads or tails, and we can treat the outcomes collectively.

2. **Define the Variables**:
- Let \( n \) be the total number of tosses. Since we are tossing 2 coins 5 times, \( n = 10 \).
- Let \( r \) be the number of heads we want to get. In this case, we want exactly 5 heads, so \( r = 5 \).
- The probability of getting heads in a single toss (for one coin) is \( p = \frac{1}{2} \) and the probability of getting tails is \( q = \frac{1}{2} \).

3. **Use the Binomial Probability Formula**:
- The probability of getting exactly \( r \) successes (heads) in \( n \) trials (tosses) is given by the formula:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
- Here, \( \binom{n}{r} \) is the binomial coefficient, which represents the number of ways to choose \( r \) successes in \( n \) trials.

4. **Substitute the Values into the Formula**:
- Substitute \( n = 10 \), \( r = 5 \), \( p = \frac{1}{2} \), and \( q = \frac{1}{2} \):
\[ P(X = 5) = \binom{10}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{10-5} \]
- This simplifies to:
\[ P(X = 5) = \binom{10}{5} \left(\frac{1}{2}\right)^{10} \]

5. **Calculate the Binomial Coefficient**:
- The binomial coefficient \( \binom{10}{5} \) can be calculated as:
\[ \binom{10}{5} = \frac{10!}{5! \cdot 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \]

6. **Combine the Results**:
- Now substitute the value of \( \binom{10}{5} \) back into the probability formula:
\[ P(X = 5) = 252 \cdot \left(\frac{1}{2}\right)^{10} = 252 \cdot \frac{1}{1024} \]

7. **Simplify the Probability**:
- This gives:
\[ P(X = 5) = \frac{252}{1024} \]
- This fraction can be simplified:
\[ P(X = 5) = \frac{63}{256} \]

### Final Answer:
The probability of getting exactly 5 heads and exactly 5 tails when two coins are tossed five times is \( \frac{63}{256} \).