The total number of positive integral solution of 15<(x1+x2+x3)<=20 is equal to a. 685 b. 785 c. 1125 d. none of these

Options

A. 1125
B. 1150
C. 1245
D. 685

Answer

To find the total number of positive integral solutions for the inequality \( 15 < (x_1 + x_2 + x_3) \leq 20 \), we can break it down into manageable steps.

### Step 1: Define the Range
We need to find the values of \( x_1 + x_2 + x_3 \) that satisfy:
\[ 15 < (x_1 + x_2 + x_3) \leq 20 \]
This means that \( x_1 + x_2 + x_3 \) can take values 16, 17, 18, 19, or 20.

### Step 2: Set Up the Equation
Let \( S = x_1 + x_2 + x_3 \). We will consider each case for \( S \) from 16 to 20.

### Step 3: Count Solutions for Each Case
We will use the stars and bars method to find the number of positive integral solutions for each value of \( S \).

1. **For \( S = 16 \)**:
\[ x_1 + x_2 + x_3 = 16 \]
We can rewrite this as:
\[ y_1 + y_2 + y_3 = 16 - 3 = 13 \]
where \( y_i = x_i - 1 \) (since \( x_i \) must be positive). The number of non-negative integral solutions is given by:
\[ \binom{13 + 3 - 1}{3 - 1} = \binom{15}{2} = \frac{15 \times 14}{2} = 105 \]

2. **For \( S = 17 \)**:
\[ x_1 + x_2 + x_3 = 17 \]
Rewriting gives:
\[ y_1 + y_2 + y_3 = 17 - 3 = 14 \]
The number of solutions is:
\[ \binom{14 + 3 - 1}{3 - 1} = \binom{16}{2} = \frac{16 \times 15}{2} = 120 \]

3. **For \( S = 18 \)**:
\[ x_1 + x_2 + x_3 = 18 \]
Rewriting gives:
\[ y_1 + y_2 + y_3 = 18 - 3 = 15 \]
The number of solutions is:
\[ \binom{15 + 3 - 1}{3 - 1} = \binom{17}{2} = \frac{17 \times 16}{2} = 136 \]

4. **For \( S = 19 \)**:
\[ x_1 + x_2 + x_3 = 19 \]
Rewriting gives:
\[ y_1 + y_2 + y_3 = 19 - 3 = 16 \]
The number of solutions is:
\[ \binom{16 + 3 - 1}{3 - 1} = \binom{18}{2} = \frac{18 \times 17}{2} = 153 \]

5. **For \( S = 20 \)**:
\[ x_1 + x_2 + x_3 = 20 \]
Rewriting gives:
\[ y_1 + y_2 + y_3 = 20 - 3 = 17 \]
The number of solutions is:
\[ \binom{17 + 3 - 1}{3 - 1} = \binom{19}{2} = \frac{19 \times 18}{2} = 171 \]

### Step 4: Sum the Solutions
Now, we sum the number of solutions for each case:
\[ 105 + 120 + 136 + 153 + 171 = 685 \]

### Final Answer
The total number of positive integral solutions of \( 15 < (x_1 + x_2 + x_3) \leq 20 \) is **685**.