Eight indentical 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. 35
B. 15
C. 140
D. 70

Answer

To solve the problem of finding the total number of ways in which the number of heads is equal to the number of tails when 8 identical coins are arranged in a row, we can follow these steps:

### Step 1: Understand the Problem
We have 8 identical coins, and we need to find arrangements where the number of heads (H) is equal to the number of tails (T). Since there are 8 coins, if the number of heads equals the number of tails, then each must be 4 (i.e., 4 heads and 4 tails).

### Step 2: Set Up the Combination Formula
The total number of arrangements of the coins can be calculated using the combination formula. The formula for combinations is given by:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

In our case, \( n = 8 \) (the total number of coins), and \( r = 4 \) (the number of heads, which is equal to the number of tails).

### Step 3: Calculate the Combinations
Using the formula, we can calculate the number of ways to choose 4 heads out of 8 coins:

\[ C(8, 4) = \frac{8!}{4! \times (8-4)!} = \frac{8!}{4! \times 4!} \]

### Step 4: Simplify the Factorials
Now, we can simplify the factorials:

\[ C(8, 4) = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4! \times 4!} \]

The \( 4! \) in the numerator and denominator cancels out:

\[ C(8, 4) = \frac{8 \times 7 \times 6 \times 5}{4!} \]

### Step 5: Calculate \( 4! \)
Now, calculate \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 6: Substitute and Calculate
Substituting \( 4! \) back into the equation:

\[ C(8, 4) = \frac{8 \times 7 \times 6 \times 5}{24} \]

Calculating the numerator:

\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]
\[ 336 \times 5 = 1680 \]

Now, divide by 24:

\[ C(8, 4) = \frac{1680}{24} = 70 \]

### Conclusion
Thus, the total number of ways in which the number of heads is equal to the number of tails when 8 identical coins are arranged in a row is **70**.