STATEMENT -1: The range of the function f(x) = `""^(7 - x)C_(x - 3)` is {3,4,5}
STATEMENT -2 : The number of squares which can be formed on a chessboard (8 `xx` 8 ) will be 204.

Options

A. (a)Statement-1 is true ,statement -2 is true , statement -2 is a correct explanation for statement -4
B. (b)Statement -1 is true , statement -2 is true , statement -2 is NOT a correct explanation for statement -4
C. (c)Statement -1 is true , statement -2 is false
D. (d)Statement -1 is false , statement -2 is true

Answer

To solve the given problem, we will analyze both statements step by step.

### Statement 1:
We need to find the range of the function \( f(x) = \binom{7 - x}{x - 3} \).

1. **Identify the conditions for the binomial coefficient**:
The binomial coefficient \( \binom{n}{r} \) is defined when \( n \geq r \) and \( r \geq 0 \). Here, we have:
- \( n = 7 - x \)
- \( r = x - 3 \)

Therefore, we need:
- \( 7 - x \geq x - 3 \)
- \( x - 3 \geq 0 \)

2. **Solve the inequalities**:
- From \( 7 - x \geq x - 3 \):
\[ 7 + 3 \geq 2x \implies 10 \geq 2x \implies x \leq 5 \]
- From \( x - 3 \geq 0 \):
\[ x \geq 3 \]

Thus, we have the range for \( x \):
\[ 3 \leq x \leq 5 \]

3. **Evaluate \( f(x) \) at the endpoints**:
- For \( x = 3 \):
\[ f(3) = \binom{7 - 3}{3 - 3} = \binom{4}{0} = 1 \]
- For \( x = 4 \):
\[ f(4) = \binom{7 - 4}{4 - 3} = \binom{3}{1} = 3 \]
- For \( x = 5 \):
\[ f(5) = \binom{7 - 5}{5 - 3} = \binom{2}{2} = 1 \]

4. **Determine the range**:
The values obtained are:
- \( f(3) = 1 \)
- \( f(4) = 3 \)
- \( f(5) = 1 \)

Therefore, the function takes the values \( 1 \) and \( 3 \) within the interval \( [3, 5] \). The range of \( f(x) \) is \( \{1, 3\} \).

### Conclusion for Statement 1:
The statement claims that the range is \( \{3, 4, 5\} \), which is incorrect. Thus, **Statement 1 is false**.

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### Statement 2:
We need to find the total number of squares that can be formed on an 8x8 chessboard.

1. **Count squares of different sizes**:
- For \( 1 \times 1 \) squares: There are \( 8 \times 8 = 64 \) squares.
- For \( 2 \times 2 \) squares: There are \( 7 \times 7 = 49 \) squares.
- For \( 3 \times 3 \) squares: There are \( 6 \times 6 = 36 \) squares.
- For \( 4 \times 4 \) squares: There are \( 5 \times 5 = 25 \) squares.
- For \( 5 \times 5 \) squares: There are \( 4 \times 4 = 16 \) squares.
- For \( 6 \times 6 \) squares: There are \( 3 \times 3 = 9 \) squares.
- For \( 7 \times 7 \) squares: There are \( 2 \times 2 = 4 \) squares.
- For \( 8 \times 8 \) squares: There is \( 1 \times 1 = 1 \) square.

2. **Sum the total number of squares**:
\[ \text{Total} = 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 \]

Calculating this:
\[ 64 + 49 = 113 \]
\[ 113 + 36 = 149 \]
\[ 149 + 25 = 174 \]
\[ 174 + 16 = 190 \]
\[ 190 + 9 = 199 \]
\[ 199 + 4 = 203 \]
\[ 203 + 1 = 204 \]

### Conclusion for Statement 2:
The total number of squares that can be formed on an 8x8 chessboard is indeed \( 204 \). Thus, **Statement 2 is true**.

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### Final Conclusion:
- Statement 1 is false.
- Statement 2 is true.