What is the greatest value of y such that 4^y is a factor of 9! ?

Since 4 = 2^2, we are actually trying to determine the largest value y such that 2^(2y) is a factor of 9!.

Let’s first determine the number of factors of 2 within 9!. To do that, we can use the following shortcut in which we divide 9 by 2, and then divide the quotient of 9/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient.

9/2 = 4 (we can ignore the remainder)

4/2 = 2

2/2 = 1

Since 1/2 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 2 within 9!.

Thus, there are 4 + 2 + 1 = 7 factors of 2 within 9!

However, we are not asked for the number of factors of 2; instead we are asked for the number of factors of 4. We see that 7 factors of 2 will produce 3 factors of 4.

Answer: C