Practice Mathematics and Probability with the exercise "Bag of Balls"

Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.

Statement

Goal

Imagine you have a bag, which is filled with N balls. Each ball can have the color black or white. You know that W balls in the bag are white.

If you extract a ball from the bag, the probability changes of the remaining balls.

Your job is it to output the ratio of the odds that you extracted k white balls after extracting s balls from the bag.
Balls are never put back into the bag!

Example:
You have 3 balls in the bag, 2 of which are white.
What is the probability of having 2 white balls by extracting 2 balls?

2/3 * 1/2 = 1/3

When A is the event of getting 2 white balls:
P(A) = Probability of getting 2 white balls = 2/3 * 1/2 = 1/3
P(!A) = Probability of not getting 2 white balls = 1-1/3 = 2/3

So the ratio P(A) : P(!A) is 1:2 (read 1 to 2).

The ratio 2:4 would be the same ratio, but both numbers are divisible by 2, so you need to reduce the ratio.

Input

Line 1: An Integer N for the number of balls in your bag.
Line 2: An Integer W for the number of white balls in your bag.
Line 3: An Integer s for the size of your sample.
Line 4: An Integer k for the desired number of white balls in your sample.

Output

Line 1: A : B, the odds ratio (in lowest terms) that k white balls are in your sample of s balls

Constraints

0 ≤ N ≤ 60
0 ≤ W ≤ N
0 ≤ s ≤ N
0 ≤ k ≤ s
0 ≤ k ≤ W

Example

Input

Output

1:2

Solve it

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Learning Opportunities

Statement

Goal

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