If eight coins are tossed together, then the probability of getting exactly 3 heads is

Options

A. `1/256`
B. `7/32`
C. `5/32`
D. `3/32`

Explanation

To find the probability of getting exactly 3 heads when tossing 8 coins, we can use the binomial probability formula. Here’s a step-by-step solution:

### Step 1: Identify the parameters
- Number of trials (n): 8 (since we are tossing 8 coins)
- Number of successes (x): 3 (we want exactly 3 heads)
- Probability of success (p): 1/2 (the probability of getting heads in a single toss)
- Probability of failure (q): 1/2 (the probability of getting tails in a single toss)

### Step 2: Use the binomial probability formula
The formula for the probability of getting exactly x successes in n trials is given by:
\[ P(X = x) = \binom{n}{x} p^x q^{n-x} \]
Where:
- \(\binom{n}{x}\) is the binomial coefficient, calculated as \(\frac{n!}{x!(n-x)!}\)

### Step 3: Calculate the binomial coefficient
For our case:
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]
Calculating this:
\[ = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

### Step 4: Calculate the probabilities
Now we need to calculate \(p^x\) and \(q^{n-x}\):
\[ p^x = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]
\[ q^{n-x} = \left(\frac{1}{2}\right)^{8-3} = \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]

### Step 5: Combine all parts to find the probability
Now we can substitute these values back into the formula:
\[ P(X = 3) = \binom{8}{3} \cdot p^3 \cdot q^{5} \]
\[ = 56 \cdot \frac{1}{8} \cdot \frac{1}{32} \]
Calculating this:
\[ = 56 \cdot \frac{1}{256} = \frac{56}{256} = \frac{7}{32} \]

### Final Answer
Thus, the probability of getting exactly 3 heads when tossing 8 coins is:
\[ \frac{7}{32} \]