The number of ways in which `5^7` can be expressed as a product of three factors:

Options

A. a. 7
B. b. 8
C. c. 9
D. d. none of these

Answer

To determine the number of ways in which \( 5^7 \) can be expressed as a product of three factors, we can follow these steps:

### Step 1: Set Up the Equation
We can express \( 5^7 \) as a product of three factors:
\[ 5^7 = 5^a \times 5^b \times 5^c \]
This implies that:
\[ a + b + c = 7 \]

### Step 2: Find Non-Negative Integer Solutions
We need to find the number of non-negative integer solutions to the equation \( a + b + c = 7 \). This can be solved using the "stars and bars" theorem in combinatorics.

### Step 3: Apply Stars and Bars Theorem
According to the stars and bars theorem, the number of ways to distribute \( n \) indistinguishable objects (stars) into \( k \) distinguishable boxes (factors) is given by:
\[ \binom{n+k-1}{k-1} \]
In our case, \( n = 7 \) (the total exponent) and \( k = 3 \) (the number of factors).

### Step 4: Calculate the Combinations
Plugging in the values:
\[ \text{Number of ways} = \binom{7 + 3 - 1}{3 - 1} = \binom{9}{2} \]
Calculating \( \binom{9}{2} \):
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 5: Conclusion
Thus, the number of ways in which \( 5^7 \) can be expressed as a product of three factors is \( 36 \).

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