A bag contains 150 balls numbered 1 through 150. Three balls are drawn and placed on the table. The probability that the balls were drawn in increasing order can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?

Question

A bag contains 150 balls numbered 1 through 150. Three balls are drawn and placed on the table. The probability that the balls were drawn in increasing order can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?

Answer 1

number of ways to draw 3 balls from 150

= 150x149x148 = 3307800

But these include triplets of the form
ABC
ACB
BAC
BCA
CAB
CBA , that is , containing the same three numbers.
Only one of these 6 would have them in the correct increasing order ,so we divide 3307800 by 6 to get 551300
so the prob of your event = 551300/3307800
= 1/6

so a/b = 1/6, and a+b = 7

Answer 2

To calculate the probability that the balls were drawn in increasing order, we need to determine the total number of possible sets of three balls that can be drawn from the bag and the number of these sets that are in increasing order.

First, let's consider the total number of sets of three balls that can be drawn from the bag. This is calculated using combinations. We can select 3 balls out of 150 without regard to their order, so the total number of sets is given by "150 choose 3" or C(150, 3), which can be calculated as:

C(150, 3) = 150! / (3! * (150 - 3)!)

Next, let's consider the number of sets in increasing order. Since the balls are numbered, there is only one way to arrange them in increasing order. So, the number of sets in increasing order is simply 1.

To calculate the probability, we divide the number of sets in increasing order by the total number of sets:

Probability = 1 / C(150, 3)

Now, let's simplify this expression to obtain the value of a/b, where a and b are coprime positive integers.

Probability = 1 / (150! / (3! * (150 - 3)!))

To simplify the expression further, we can use the fact that (n - r)! = n! / r!.

Probability = (3! * (150 - 3)!) / 150!

Now, let's simplify the factorials:

Probability = (3 * 2 * 1 * (150 - 3)! / (150 * 149 * 148 * ... * 4 * 3 * 2 * 1)

Canceling out some terms, we can simplify this to:

Probability = (3 * 2 * 1) / (150 * 149 * 148)

Finally, we can write this probability as a/b, where a = 3 * 2 * 1 and b = 150 * 149 * 148.

a = 3 * 2 * 1 = 6
b = 150 * 149 * 148

Now, we can calculate the value of a + b:

a + b = 6 + (150 * 149 * 148)