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

Options

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

Answer

To solve the problem of finding the total number of ways to arrange 6 identical coins such that the number of heads is equal to the number of tails, we can follow these steps:

### Step 1: Understand the Problem
We have 6 identical coins, and we need to find the arrangements where the number of heads (H) is equal to the number of tails (T). Since there are 6 coins, for heads to equal tails, we need 3 heads and 3 tails.

### Step 2: Determine the Required Arrangement
Since we need 3 heads and 3 tails, we can represent this as choosing 3 positions for heads out of the 6 available positions. The remaining positions will automatically be filled with tails.

### Step 3: Use Combinations to Calculate the Arrangements
The number of ways to choose 3 positions from 6 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 6 \) and \( r = 3 \):
\[ \binom{6}{3} = \frac{6!}{3! \cdot (6-3)!} = \frac{6!}{3! \cdot 3!} \]

### Step 4: Calculate Factorials
Now, we calculate the factorials:
- \( 6! = 720 \)
- \( 3! = 6 \)

Substituting these values into the combination formula:
\[ \binom{6}{3} = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20 \]

### Step 5: Conclusion
Thus, the total number of ways to arrange the coins such that the number of heads is equal to the number of tails is **20**.

### Final Answer
The total number of ways in which the number of heads is equal to the number of tails is **20**.

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