Statement-1 If 10 coins are thrown simultaneously, then the probability of appearing exactly four heads is equal to probability of appearing exactly six heads.
Statement-2 `.^nC_r=.^nC_s implies` either r=s or r+s=n and P(H)=P(T) in a single trial.
(a)Statement-1 is true, Statement-2 is true: Statement-2 is a correct explanation for Statement-1
(b)Statement-1 is true, Statement-2 is true: Statement-2 is not a correct explanation for Statement-1
(c)Statement-1 is true, Statement-2 is false
(d)Statement-1 is false, Statement-2 is true

Options

A. Statement-1 is true, Statement-2 is true: Statement-2 is a correct explanation for Statement-1
B. Statement-1 is true, Statement-2 is true: Statement-2 is not a correct explanation for Statement-1
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

To solve the problem, we will analyze both statements and determine their validity step by step.

### Step 1: Analyze Statement 1
Statement 1 claims that if 10 coins are thrown simultaneously, the probability of getting exactly 4 heads is equal to the probability of getting exactly 6 heads.

1. **Understanding the Coin Toss**:
- Each coin has two outcomes: Heads (H) or Tails (T).
- The probability of getting Heads (P(H)) = 1/2 and the probability of getting Tails (P(T)) = 1/2.

2. **Using Binomial Probability Formula**:
- The probability of getting exactly r heads in n tosses is given by the formula:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
- Here, \( n = 10 \), \( p = 1/2 \).

3. **Calculate Probability for 4 Heads**:
\[ P(X = 4) = \binom{10}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{10-4} = \binom{10}{4} \left(\frac{1}{2}\right)^{10} \]

4. **Calculate Probability for 6 Heads**:
\[ P(X = 6) = \binom{10}{6} \left(\frac{1}{2}\right)^6 \left(\frac{1}{2}\right)^{10-6} = \binom{10}{6} \left(\frac{1}{2}\right)^{10} \]

5. **Using the Property of Binomial Coefficients**:
- We know that \( \binom{n}{r} = \binom{n}{n-r} \).
- Therefore, \( \binom{10}{6} = \binom{10}{4} \).

6. **Conclusion for Statement 1**:
- Since \( P(X = 4) = P(X = 6) \), Statement 1 is true.

### Step 2: Analyze Statement 2
Statement 2 states that \( \binom{n}{r} = \binom{n}{s} \) implies either \( r = s \) or \( r + s = n \), and that \( P(H) = P(T) \) in a single trial.

1. **Understanding the Binomial Coefficient Property**:
- The property \( \binom{n}{r} = \binom{n}{s} \) holds true if:
- \( r = s \) (they are the same) or
- \( r + s = n \) (they are complementary).

2. **Probability Condition**:
- The statement also mentions \( P(H) = P(T) \), which is true for a fair coin.

3. **Conclusion for Statement 2**:
- Both parts of Statement 2 are true.

### Final Conclusion
- Since Statement 1 is true and Statement 2 is also true, and Statement 2 correctly explains Statement 1, the correct option is:
**(a) Statement-1 is true, Statement-2 is true: Statement-2 is a correct explanation for Statement-1.**