Check out this solution to Monday’s Get Your Factorials Straight! Brain Twister!

## Question

__Column A__16!^100 – 9!^100

__Column B__

(4!^100 – 3!^100)(4!^100 + 3!^100)

## Answer and Explanation

It is tempting to think that the two columns are equal: x^2 – y^2 = (x – y)(x + y).

But 4! x 4! does not equal 16! Compare below:

16! = 16x15x 14×13…x2x 1

4! x 4! = 4x3x2x1x4x3x2x1

You’ll notice that 16! Is much greater than 4!. Therefore, we can conclude that Column A is going to be much bigger. True, we are subtracting 9! from 16!. But even then, when we factor the quantity in column A, we get 9!(16x15x14x13x12x11x10 – 1). The part on the left—where we are multiplying everything up is still going to be much greater than 4!x4!, which is only 576.

You may have noticed that I’ve totally omitted mention of the fact that there are massive to the power of 100 next to each of the quantities mentioned above. The thing is we can discount these, since they are equal for each quantity. In other words, when we know that a given positive integer greater than 1, say ‘x’, is bigger than another positive integer, say ‘y’, that x to any positive power, regardless of how large, is going to be larger than ‘y’ to that same power.

That leaves me with answer (A).

Hi,

Doesn’t Column B (4!^100 – 3!^100)(4!^100 + 3!^100) simplify to 4!^200 – 9!^200? If it is then doesn’t this preclude us from discounting the exponents when comparing with Column A 16!^100 – 9!^100? Thanks and I’d appreciate a clarification.

Hi Meg 🙂

You’re on the right track, but it looks like you made a small mistake when multiplying out (4!^100 – 3!^100)(4!^100 + 3!^100):

(4!^100 – 3!^100)(4!^100 + 3!^100)

= (4!^100)(4!^100) + (4!^100)(3!^100) – (4!^100)(3!^100) – (3!^100)(3!^100)

= (4!^100)(4!^100) – (3!^100)(3!^100)

= 4!^200 –

3!^200Just as you did for the product (4!^100)(4!^100), we need to add the exponents together to simplify the second term, not square the base (3!):

(3!^100)(3!^100) = 3!^200

not9!^200For that reason, Chris approached the question using the method explained above 🙂

Hope this clears up your doubts!