Question

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

• A. 7
• B. 8
• C. 9
• D. none of these

Answer 1

The correct option is B 8

$5^7=5^a\times 5^b\times 5^c$
Such that a+b+c=7 where a, b, c are all whole numbers.
Now we know that a+b+c=7 has ${}^{7+3-1}{C}_2{=}^9{C}_2=36$ number of solutions.
But 36 incluces those solutions which have any 2 variables equal to each other. Let's check for those. Let us consider a= b
then 2a + c = 7
For the above equation, the number of solutions is 4. But since any 2 variables can be taken therefore total number of such cases = ${}^3{C}_2\times 4=12$
Removing these cases and keeping in mind that we have to find out the number of ways in an unordered form, we get $\frac{36-12}{6}+4=4+4=8$
Hence, total number of ways = 8
$\{0,0,7\},\{0,4,3\},\{0,6,1\},\{0,5,2\},\{1,4,2\},\{1,3,3\},\{1,1,5\},\{2,2,3\}$