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 solve the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 101 \) where \( x_i \) are odd natural numbers, we can follow these steps:

### Step 1: Express odd natural numbers
Since \( x_i \) are odd natural numbers, we can express each \( x_i \) in the form:
\[ x_i = 2n_i + 1 \]
where \( n_i \) is a non-negative integer (i.e., \( n_i \geq 0 \)).

### Step 2: Substitute into the equation
Substituting the expression for \( x_i \) into the 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:
\[ 2n_1 + 2n_2 + 2n_3 + 2n_4 + 2n_5 + 5 = 101 \]
Subtracting 5 from both sides results in:
\[ 2n_1 + 2n_2 + 2n_3 + 2n_4 + 2n_5 = 96 \]

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

### Step 5: Count the solutions
Now we need to find the number of non-negative integer solutions to the equation \( n_1 + n_2 + n_3 + n_4 + n_5 = 48 \). This is a classic problem that can be solved using the "stars and bars" theorem.

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 to be divided (48 in this case) and \( r \) is the number of variables (5 in this case).

### Step 6: Apply the formula
Substituting \( n = 48 \) and \( r = 5 \) into the formula gives:
\[ \binom{48 + 5 - 1}{5 - 1} = \binom{52}{4} \]

### Conclusion
Thus, the number of solutions to the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 101 \) where \( x_i \) are odd natural numbers is:
\[ \binom{52}{4} \]