If five coins are flipped and all the different ways they could fall are listed, how many elements of this list will contain more than 3 heads?

Options

A. 5
B. 6
C. 10
D. 16

Explanation

To solve the problem of finding how many ways five coins can be flipped to show more than three heads, we will consider the cases of getting exactly four heads and exactly five heads.

### Step-by-Step Solution:

1. **Identify the Total Number of Coins**:
We have a total of 5 coins.

2. **Define the Cases for More Than 3 Heads**:
- More than 3 heads means we can have either 4 heads or 5 heads.

3. **Calculate the Number of Ways to Get 4 Heads**:
- The number of ways to choose 4 heads from 5 coins can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items (coins) and \( r \) is the number of items to choose (heads).
- Here, \( n = 5 \) and \( r = 4 \).
- The formula for combinations is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Applying the values:
\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} = \frac{5 \times 4!}{4! \cdot 1} = 5 \]

4. **Calculate the Number of Ways to Get 5 Heads**:
- The number of ways to choose 5 heads from 5 coins is:
- Here, \( n = 5 \) and \( r = 5 \).
- Applying the combination formula:
\[ \binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{5!}{5! \cdot 0!} = \frac{5!}{5! \cdot 1} = 1 \]

5. **Sum the Results**:
- Now, we add the number of ways to get 4 heads and 5 heads:
\[ \text{Total} = \binom{5}{4} + \binom{5}{5} = 5 + 1 = 6 \]

6. **Conclusion**:
- Therefore, the total number of ways to flip 5 coins such that there are more than 3 heads is **6**.