Find the number of positive integral solutions satisfying the equation `(x_1+x_2+x_3)(y_1+y_2)=77.`

Explanation

To find the number of positive integral solutions satisfying the equation \((x_1 + x_2 + x_3)(y_1 + y_2) = 77\), we can follow these steps:

### Step 1: Factor the Equation
We start by expressing 77 as a product of two factors. The possible pairs of factors of 77 are:
- \(1 \times 77\)
- \(7 \times 11\)
- \(11 \times 7\)
- \(77 \times 1\)

### Step 2: Analyze the Factors
We will analyze the pairs of factors that can yield positive integral solutions for \(x_1 + x_2 + x_3\) and \(y_1 + y_2\).

1. **Case 1**: \(x_1 + x_2 + x_3 = 7\) and \(y_1 + y_2 = 11\)
2. **Case 2**: \(x_1 + x_2 + x_3 = 11\) and \(y_1 + y_2 = 7\)

The cases \(x_1 + x_2 + x_3 = 1\) or \(y_1 + y_2 = 1\) are not valid since we need positive integers.

### Step 3: Calculate Solutions for Case 1
For \(x_1 + x_2 + x_3 = 7\):
- The number of positive integral solutions can be found using the formula for combinations:
\[ \text{Number of solutions} = \binom{n-1}{r-1} = \binom{7-1}{3-1} = \binom{6}{2} = 15 \]

For \(y_1 + y_2 = 11\):
\[ \text{Number of solutions} = \binom{11-1}{2-1} = \binom{10}{1} = 10 \]

Thus, the total number of solutions for Case 1 is:
\[ 15 \times 10 = 150 \]

### Step 4: Calculate Solutions for Case 2
For \(x_1 + x_2 + x_3 = 11\):
\[ \text{Number of solutions} = \binom{11-1}{3-1} = \binom{10}{2} = 45 \]

For \(y_1 + y_2 = 7\):
\[ \text{Number of solutions} = \binom{7-1}{2-1} = \binom{6}{1} = 6 \]

Thus, the total number of solutions for Case 2 is:
\[ 45 \times 6 = 270 \]

### Step 5: Add the Solutions from Both Cases
Now, we add the solutions from both cases:
\[ 150 + 270 = 420 \]

### Conclusion
The total number of positive integral solutions satisfying the equation \((x_1 + x_2 + x_3)(y_1 + y_2) = 77\) is **420**.

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