Compute the number of positive integer divisors of 10

[RM86Z]

Homework Statement

number of positive integer divisors of 10!

Relevant Equations

10!

Compute the number of positive integer divisors of 10!. By the fundamental theorem of arithmetic and the factorial expansion:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
= 2 x 5 x 3^2 x 7 x 2 x 3 x 5 x 2^2 x 3 x 2 x 1
= 2^8 x 3^4 x 5^2 x 7

Then there are 9 possibilities for 2, 5 for 3, 3 for 5 and for 7 giving 9 x 5 x 3 = 135.

The book gives 270 as the answer, where am I going wrong?

Thank you!

EDIT:Oops, I should have counted 7 as two giving 9 x 5 x 3 x 2 = 270!

 

[fresh_42]

You can find a formula to cross-check on Wikipedia:

https://en.wikipedia.org/wiki/Divisor_function

 

[haruspex]

Homework Statement:: number of positive integer divisors of 10!
Relevant Equations:: 10!

270!

Isn't that greater than 10! ?