# Thread: Set numbers

1. ## Set numbers

A number a is to be chosen at random from the set of {1,2,3,4,5,6}. A number b is to be chosen at random from the remaining five numbers. What is the probability that a/b will be an integer?

How can I prove that the answer is indeed 4/15?

2. ## Re: Set numbers

Originally Posted by donnagirl
A number a is to be chosen at random from the set of {1,2,3,4,5,6}. A number b is to be chosen at random from the remaining five numbers. What is the probability that a/b will be an integer?
How can I prove that the answer is indeed 4/15?
Well there thirty pairs $(a,b)$ from the cross product such that $a\ne b$.
You can list the pairs $(a,b)$ such that $\frac{a}{b}\in \mathbb{N}.$ Then count them.

3. ## Re: Set numbers

Is there more of a proof way or combinatorial way to approach then just brute force counting?

4. ## Re: Set numbers

Originally Posted by donnagirl
Is there more of a proof way or combinatorial way to approach then just brute force counting?
I don't what to say that. With such a small outcome what would anyone want to do it the hard way?
Let $d_n$ be the number of divisors of $n$ less than $n$.
$d_1=0,~d_2=1,~d_3=1,~d_4=2,~d_5=1,~d_6=3$
There are eight such pairs out of thirty pairs.

If your set were $\{1,2,3,\cdots,98,99\}$ then the answer is: $\frac{{\sum\limits_{n = 2}^{99} {{d_n}} }}{{(99)(98)}}$

5. ## Re: Set numbers

I am not sure if this simplifies the counting, but you can select b first and then a. This does not change the answer. For b = 1 there are 5 multiples of b among {1, 2, 3, 4, 5, 6} - {b}, for b = 2 there are 2 multiples, and for b = 3 there is 1 multiple. For other b's there are no multiples distinct from b in this set. The number of "successful" pairs (a, b) is 5 + 2 + 1 = 8.

6. ## Re: Set numbers

You may have to just list the ordered pairs (a,b) based on the value of b:

(2,1) (3,1) (4,1) (5,1) (6,1)
(4,2) (6,2)
(6,3)

There are 6P2 = 30 ways to select a,b and 8 of them work. The probability is 8/30 = 4/15.

7. ## Re: Set numbers

Code:
a:   1   2   3   4   5    6

b:   0   1   1  1,2  1  1,2,3  : 8
5   5   5   5   5    5    :30