# Question: Total Number of Balls

## Comment on Total Number of Balls

it is easy

### I feel kind of silly, but I

I feel kind of silly, but I reduced the fractions.

So I got 1/12 = 6/(n)(n-1)

I reduced and got n²-n = 2.

(n+1)(n-2) = 0

S0 n = -1,2

I guess I shouldn't have reduced, right?

But aren't we always taught to reduce?

So, knowing I did reduce, would 2 be a wrong answer?

Could you please tell me why I was wrong to reduce here? I feel like I should know why, but I don't really.

### The reducing you performed is

The reducing you performed is the kind of reducing might do before MULTIPLYING fractions. In your solution above, you are not multiplying fractions.
For example, before multiplying 3/50 x 20/21, we can reduce everything to 1/5 x 2/7
HOWEVER, this only helps make it easier to MULTIPLY fractions.
It doesn't create equivalent fractions.
For example, in the above example, we're not saying that 3/50 is equivalent to 1/5
We're just recognizing that performing this step will make our calculations easier.
For more on this, see https://www.greenlighttestprep.com/module/gre-arithmetic/video/1069

If the actual equation were n²-n = 2, then your solution (n = -1 or 2) would be perfect.

Cheers,
Brent

### So, I am a bit intrigued. Why

So, I am a bit intrigued. Why can't we proceed in this way: Selecting two white balls = 2/NC3=1/12?
And, then solving for N?

Thank you Brent!

### This question follows the

This question follows the lesson on rewriting probabilities so that we can apply PROBABILITY rules, so the solution follows that strategy.

However, as with many probability questions, we can apply COUNTING rules.
In this case, your equation, 2/NC3 = 1/12 isn't quite right.

P(both balls are white) = (number of ways to get 2 white balls)/(TOTAL number of ways to select ANY 2 balls)

Number of ways to get 2 white balls:
There are 3 white balls in total, so we can select 2 white balls in 3C2 ways
3C2 = 3

TOTAL number of ways to select ANY 2 balls:
There are N balls in total, so we can any 2 balls in NC2 ways

So, our equation becomes: 3/(NC2) = 1/12

Since NC2 = N(N - 1)/2, we can rewrite our equation as: 3/[N(N - 1)/2] = 1/12
Simplify to get: (3)[2/N(N - 1)] = 1/12
Simplify: 6/N(N - 1) = 1/12
Cross multiply to get: N(N - 1) = (6)(12)
Simplify to get: N² - N = 72
Rewrite as: N² - N - 72 = 0
Factor: (N + 8)(N - 9) - 0

So, EITHER N = -8 OR N = 9
Since N cannot be negative, we can conclude that N = 9

Cheers,
Brent