The number of solutions of the equation ` x_(1)+x_(2) + x_(3) + x_(4) + x_(5) = 101`, where `x_(i)^(') s` are odd natural numbers is :

Options

A. ` ""^(105)C_(4)`
B. ` ""^(52)C_(5)`
C. ` ""^(52)C_(4)`
D. ` ""^(50)C_(4)`

Answer

To find the number of solutions for the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 101 \) where each \( x_i \) is an odd natural number, we can follow these steps:

### Step 1: Represent Odd Natural Numbers
Each \( x_i \) (where \( i = 1, 2, 3, 4, 5 \)) can be expressed as an odd natural number. The general form for an odd natural number can be written as:
\[ x_i = 2n_i + 1 \]
where \( n_i \) is a non-negative integer.

### Step 2: Substitute into the Equation
Substituting \( x_i \) into the original equation gives:
\[ (2n_1 + 1) + (2n_2 + 1) + (2n_3 + 1) + (2n_4 + 1) + (2n_5 + 1) = 101 \]

### Step 3: Simplify the Equation
This simplifies to:
\[ 2(n_1 + n_2 + n_3 + n_4 + n_5) + 5 = 101 \]
Subtracting 5 from both sides, we have:
\[ 2(n_1 + n_2 + n_3 + n_4 + n_5) = 96 \]

### Step 4: Divide by 2
Dividing both sides by 2 gives:
\[ n_1 + n_2 + n_3 + n_4 + n_5 = 48 \]

### Step 5: Use the Stars and Bars Theorem
Now, we need to find the number of non-negative integer solutions to the equation \( n_1 + n_2 + n_3 + n_4 + n_5 = 48 \). According to the stars and bars theorem, the number of solutions is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number (48) and \( r \) is the number of variables (5).

### Step 6: Calculate the Combination
In our case:
\[ n = 48 \quad \text{and} \quad r = 5 \]
Thus, we need to calculate:
\[ \binom{48 + 5 - 1}{5 - 1} = \binom{52}{4} \]

### Step 7: Final Calculation
Now, we compute \( \binom{52}{4} \):
\[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \]

### Conclusion
The number of solutions to the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 101 \) where each \( x_i \) is an odd natural number is \( 270725 \).