Six identical coins are arranged in a row. The total number of ways in which the number of heads is equal to the number of tails is (a) 9 (b) 20 (c) 40 (d) 120

Options

A. 9
B. 20
C. 40
D. 120

Answer

To solve the problem of arranging six identical coins such that the number of heads is equal to the number of tails, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 6 identical coins, and we need to arrange them such that there are an equal number of heads and tails. Since there are 6 coins, we can have 3 heads (H) and 3 tails (T).

2. **Identifying the Arrangement**:
The arrangement we are looking for is a combination of 3 heads and 3 tails. This means we need to find the number of ways to choose 3 positions for heads out of 6 total positions.

3. **Using Combinations**:
The number of ways to choose 3 positions from 6 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 6 \) and \( r = 3 \).

4. **Calculating the Combinations**:
Plugging in the values into the combination formula:
\[ \binom{6}{3} = \frac{6!}{3! \cdot 3!} \]

5. **Simplifying the Factorials**:
We can simplify this as follows:
\[ 6! = 6 \times 5 \times 4 \times 3! \quad \text{(since } 3! \text{ will cancel out)} \]
Therefore:
\[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} \]

6. **Calculating the Final Value**:
Now, calculating the above expression:
\[ = \frac{120}{6} = 20 \]

7. **Conclusion**:
Thus, the total number of ways in which the number of heads is equal to the number of tails is **20**. Therefore, the answer is (b) 20.