If the total number of positive integral solution of `15ltx_(1)+x_(2)+x_(3)le20` is k, then the value of `(k)/(100)` is equal to

Answer

To solve the problem of finding the total number of positive integral solutions of the inequality \(15 < x_1 + x_2 + x_3 \leq 20\), we can break it down into several steps:

### Step 1: Understand the Inequality
The inequality \(15 < x_1 + x_2 + x_3 \leq 20\) implies that the sum \(x_1 + x_2 + x_3\) can take values from 16 to 20. Therefore, we can express this as:
- \(x_1 + x_2 + x_3 = 16\)
- \(x_1 + x_2 + x_3 = 17\)
- \(x_1 + x_2 + x_3 = 18\)
- \(x_1 + x_2 + x_3 = 19\)
- \(x_1 + x_2 + x_3 = 20\)

### Step 2: Change of Variables
Since \(x_1\), \(x_2\), and \(x_3\) are positive integers, we can make a change of variables to simplify our calculations. Let:
- \(y_1 = x_1 - 1\)
- \(y_2 = x_2 - 1\)
- \(y_3 = x_3 - 1\)

This means \(y_1\), \(y_2\), and \(y_3\) are non-negative integers (i.e., \(y_i \geq 0\)). The new equation becomes:
\[ y_1 + y_2 + y_3 = n - 3 \]
where \(n\) is the sum \(x_1 + x_2 + x_3\). Thus, we can rewrite the equations for \(n = 16, 17, 18, 19, 20\) as:
- For \(n = 16\): \(y_1 + y_2 + y_3 = 16 - 3 = 13\)
- For \(n = 17\): \(y_1 + y_2 + y_3 = 17 - 3 = 14\)
- For \(n = 18\): \(y_1 + y_2 + y_3 = 18 - 3 = 15\)
- For \(n = 19\): \(y_1 + y_2 + y_3 = 19 - 3 = 16\)
- For \(n = 20\): \(y_1 + y_2 + y_3 = 20 - 3 = 17\)

### Step 3: Count the Non-negative Solutions
The number of non-negative integer solutions to the equation \(y_1 + y_2 + y_3 = k\) is given by the formula:
\[ \binom{k + r - 1}{r - 1} \]
where \(r\) is the number of variables (in this case, 3).

Thus, we can compute the number of solutions for each case:
- For \(n = 16\): \(y_1 + y_2 + y_3 = 13\) gives \(\binom{13 + 3 - 1}{3 - 1} = \binom{15}{2}\)
- For \(n = 17\): \(y_1 + y_2 + y_3 = 14\) gives \(\binom{14 + 3 - 1}{3 - 1} = \binom{16}{2}\)
- For \(n = 18\): \(y_1 + y_2 + y_3 = 15\) gives \(\binom{15 + 3 - 1}{3 - 1} = \binom{17}{2}\)
- For \(n = 19\): \(y_1 + y_2 + y_3 = 16\) gives \(\binom{16 + 3 - 1}{3 - 1} = \binom{18}{2}\)
- For \(n = 20\): \(y_1 + y_2 + y_3 = 17\) gives \(\binom{17 + 3 - 1}{3 - 1} = \binom{19}{2}\)

### Step 4: Calculate Each Binomial Coefficient
Now we calculate each binomial coefficient:
- \(\binom{15}{2} = \frac{15 \times 14}{2} = 105\)
- \(\binom{16}{2} = \frac{16 \times 15}{2} = 120\)
- \(\binom{17}{2} = \frac{17 \times 16}{2} = 136\)
- \(\binom{18}{2} = \frac{18 \times 17}{2} = 153\)
- \(\binom{19}{2} = \frac{19 \times 18}{2} = 171\)

### Step 5: Sum the Solutions
Now we sum all the solutions:
\[ k = 105 + 120 + 136 + 153 + 171 = 685 \]

### Step 6: Calculate \(\frac{k}{100}\)
Finally, we need to find \(\frac{k}{100}\):
\[ \frac{685}{100} = 6.85 \]

### Final Answer
Thus, the value of \(\frac{k}{100}\) is \(6.85\).

---