10 coins are tossed . What is the probability that exactly 5 heads appear ?

Options

A. `(63)/(256)`
B. `(126)/(256)`
C. `(186)/(256)`
D. `(65)/(256)`

Answer

To find the probability of getting exactly 5 heads when tossing 10 coins, we can follow these steps:

### Step 1: Determine the total number of outcomes when tossing 10 coins.
When a coin is tossed, there are 2 possible outcomes: heads (H) or tails (T). Therefore, when 10 coins are tossed, the total number of outcomes is given by:

\[ \text{Total Outcomes} = 2^{10} = 1024 \]

### Step 2: Calculate the number of favorable outcomes for getting exactly 5 heads.
To find the number of ways to get exactly 5 heads out of 10 coins, we can use the combination formula \( nCr \), which is defined as:

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

In our case, \( n = 10 \) (the total number of coins) and \( r = 5 \) (the number of heads we want). Thus, we need to calculate:

\[ 10C5 = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} \]

### Step 3: Simplify the combination calculation.
Expanding \( 10! \):

\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5! \]

Now substituting this back into the combination formula:

\[ 10C5 = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5! \times 5!} \]

The \( 5! \) in the numerator and denominator cancels out:

\[ 10C5 = \frac{10 \times 9 \times 8 \times 7 \times 6}{5!} \]

Calculating \( 5! \):

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Now substituting this value back:

\[ 10C5 = \frac{10 \times 9 \times 8 \times 7 \times 6}{120} \]

### Step 4: Calculate the numerator.
Calculating \( 10 \times 9 \times 8 \times 7 \times 6 \):

\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]
\[ 5040 \times 6 = 30240 \]

So, we have:

\[ 10C5 = \frac{30240}{120} \]

### Step 5: Perform the division.
Calculating \( \frac{30240}{120} \):

\[ 30240 \div 120 = 252 \]

Thus, the number of favorable outcomes (getting exactly 5 heads) is 252.

### Step 6: Calculate the probability.
The probability \( P \) of getting exactly 5 heads is given by the ratio of favorable outcomes to total outcomes:

\[ P(\text{exactly 5 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{252}{1024} \]

### Step 7: Simplify the probability.
To simplify \( \frac{252}{1024} \):

Both numbers can be divided by 4:

\[ \frac{252 \div 4}{1024 \div 4} = \frac{63}{256} \]

### Final Answer:
Thus, the probability of getting exactly 5 heads when tossing 10 coins is:

\[ \frac{63}{256} \]

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