The number of positive integral solutions of the equation `x_1 x_2 x_3 x_4 x_5 = 1050` is (A) 1800 (B) 1600 (C) 1400 (D) None of these

Options

A. 1800
B. 1600
C. 1400
D. none

Answer

To find the number of positive integral solutions of the equation \( x_1 x_2 x_3 x_4 x_5 = 1050 \), we will follow these steps:

### Step 1: Factorize 1050
First, we need to factorize 1050 into its prime factors.

\[ 1050 = 2 \times 3 \times 5^2 \times 7 \]

### Step 2: Identify the prime factors
From the factorization, we can see that the prime factors of 1050 are:
- \( 2^1 \)
- \( 3^1 \)
- \( 5^2 \)
- \( 7^1 \)

### Step 3: Distribute the prime factors among the variables
We need to distribute these prime factors among the five variables \( x_1, x_2, x_3, x_4, x_5 \).

#### For \( 2^1 \):
The factor \( 2 \) can be assigned to any of the 5 variables. Therefore, there are \( \binom{5}{1} = 5 \) ways to assign the factor of \( 2 \).

#### For \( 3^1 \):
Similarly, the factor \( 3 \) can also be assigned to any of the 5 variables. There are \( \binom{5}{1} = 5 \) ways to assign the factor of \( 3 \).

#### For \( 5^2 \):
The factor \( 5^2 \) can be assigned to two variables. We can choose 2 out of the 5 variables to receive the factor \( 5 \). The number of ways to choose 2 variables from 5 is given by \( \binom{5}{2} \).

\[ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \]

#### For \( 7^1 \):
The factor \( 7 \) can also be assigned to any of the 5 variables. There are \( \binom{5}{1} = 5 \) ways to assign the factor of \( 7 \).

### Step 4: Calculate the total number of distributions
Now, we can multiply the number of ways to assign each of the prime factors:

\[ \text{Total ways} = \binom{5}{1} \times \binom{5}{1} \times \binom{5}{2} \times \binom{5}{1} \]

Substituting the values we calculated:

\[ \text{Total ways} = 5 \times 5 \times 10 \times 5 \]

Calculating this gives:

\[ 5 \times 5 = 25 \]
\[ 25 \times 10 = 250 \]
\[ 250 \times 5 = 1250 \]

### Step 5: Conclusion
Thus, the number of positive integral solutions of the equation \( x_1 x_2 x_3 x_4 x_5 = 1050 \) is \( 1250 \).

Since this value does not match any of the provided options, the answer is (D) None of these.

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