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 are coprime positive integers. What is the value of a+b
To find the probability that the balls were drawn in increasing order, we need to consider the possible outcomes when drawing the balls.
First, the total number of possible outcomes when drawing three balls from a bag of 150 balls is given by the combination formula: C(150, 3) = (150!)/(3!(150-3)!).
Next, let's consider the cases when the balls are drawn in increasing order:
Case 1: The three balls are consecutive numbers.
There are 148 possible outcomes for this case since we can select any number from 1 to 148 as the first ball, and the next two balls will be the consecutive numbers following the first ball.
Case 2: The three balls are non-consecutive numbers.
In this case, we need to select any three numbers from the set of 150 balls, excluding the 148 possibilities from Case 1.
The number of ways to select three non-consecutive numbers is given by C(150, 3) - 148.
Therefore, the probability of drawing the balls in increasing order is:
P(increasing order) = (148 + (C(150, 3) - 148)) / C(150, 3).
Now, we can calculate the value of a/b.
To simplify the expression, we can find the common denominator, which is C(150, 3):
148 + (C(150, 3) - 148) = 148 + C(150, 3) - 148
= C(150, 3)
So, a = C(150, 3) and b = C(150, 3).
Finally, we need to find the value of a + b:
a + b = C(150, 3) + C(150, 3)
= 2 * C(150, 3)
At this point, we need to calculate the value of C(150, 3):
C(150, 3) = (150!)/(3!(150-3)!)
= (150 * 149 * 148)/(3 * 2 * 1)
= 55 * 7452
= 409,860
Therefore, a + b = 2 * 409,860 = 819,720.
Hence, the value of a + b is 819,720.