8 coins are tossed simultaneously . The probability of getting 6 head is

Options

A. `(7)/(64)`
B. `(57)/(64)`
C. `(37)/(256)`
D. `(249)/(256)`

Explanation

To find the probability of getting exactly 6 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 (r): 6 (we want to find the probability of getting 6 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 binomial probability formula is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
Where:
- \(\binom{n}{r}\) is the binomial coefficient, calculated as \(\frac{n!}{r!(n-r)!}\)

### Step 3: Calculate the binomial coefficient
Calculate \(\binom{8}{6}\):
\[ \binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8!}{6! \cdot 2!} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 4: Calculate \(p^r\) and \(q^{n-r}\)
Calculate \(p^r\) and \(q^{n-r}\):
- \(p^r = \left(\frac{1}{2}\right)^6 = \frac{1}{64}\)
- \(q^{n-r} = \left(\frac{1}{2}\right)^{8-6} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)

### Step 5: Substitute values into the formula
Now substitute the values into the binomial probability formula:
\[ P(X = 6) = \binom{8}{6} \cdot p^6 \cdot q^{2} = 28 \cdot \frac{1}{64} \cdot \frac{1}{4} \]

### Step 6: Simplify the expression
Now simplify the expression:
\[ P(X = 6) = 28 \cdot \frac{1}{64} \cdot \frac{1}{4} = 28 \cdot \frac{1}{256} = \frac{28}{256} = \frac{7}{64} \]

### Final Answer
The probability of getting exactly 6 heads when tossing 8 coins is \(\frac{7}{64}\).

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