By "without actually computing the whole thing", I'm going to assume you mean "without actually computing factorial in the traditional, brute force, way", i.e. "show any possible shortcut".... let's take a look at 25! because some of the paper-and-pencil approaches will apply to higher numbers as well...
First let's write out the factors of the numbers from 2 to 25: 2,3,2*2,5,2*3,7,2*2*2,3*3,2*5,11,2*2*3,13,2*7,3*5,2*2*2*2,17,2*3*3,19,2*2*5,3*7,2*11,23,2*2*2*3,5*5
Counting up the factors, we have 22 2s, ten 3s, etc, that is, 25! can be written as: 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23
As joriki says, we can toss tens, so let's toss out 6 5s and 6 2s:
2^16 * 3^10 * 7^3 * 11^2 * 13 * 17 * 19 * 23
We can make it still easier on ourselves by factoring out some numbers that are easier for us to work with on paper. For example, it's easy to multiply by 1001, and 1001 is 7*11*13, so let's separate those 3 factors out:
2^16 * 3^10 * 7^2 * 11 * 17 * 19 * 23 * 1001
Likewise 99 is easy to work with, so let's get that out:
2^16 * 3^8 * 7^2 * 17 * 19 * 23 * 1001 * 99
And 98 is only a little harder than 99 is:
2^15 * 3^8 * 17 * 19 * 23 * 1001 * 99 * 98
And finally, 102 is also just two away from 100:
2^14 * 3^7 * 19 * 23 * 1001 * 99 * 98 * 102
Come to think of it, 98 * 102 is 9,996, just four away from 10,000:
2^14 * 3^7 * 19 * 23 * 1001 * 99 * 9,996
And multiplying by 9 is easier than 3 so:
2^14 * 3 * 9^3 * 19 * 23 * 1001 * 99 * 9,996
We're out of tricks at this point, so let's go for it. Let's do the big primes now: 19 * 23 = 437.
Then multiply by 9 3 times: 4370 - 437 = 3933 39330 - 3933 = 35397 353970 - 35397 = 318573
And multiply by 3 once:
318573 * 3 = 955719
Then double this 14 times. (Hope I'm not the only one who finds it easy to double numbers?)
1911438 3822876 7645752 15291504 30583008 61166016 122332032 244664064 489328128 978656256 1957312512 3914625024 7829250048 15658500096
The only factors we have left are 1001, 99, and 9,996. Let's do that last one first... it's 10,000-4, so we'll double two more times:
31317000192 62634000384
And then subtract:
156585000960000 - 62634000384 =156522366959616
Multiply by 99:
15652236695961600 - 156522366959616 =15495714329001984
And finally multiply by 1001:
15495714329001984000 + 15495714329001984 =15511210043330985984
Finally, you can add up those digits.
1+5+5+1+1+2+1+0+0+4+3+3+3+0+9+8+5+9+8+4=72
as mt_ mentions, we expect the result to be evenly divisible by 9, so 72 sounds quite plausible.
p.s. Note to the stackexchange sysops: the trend on this question seems to be to post comments (not answers) to the original question. I would gladly do that, but since I still have a low rep, I am barred from posting comments.