Five marbles are drawn from a bag which contains 6 blue marbles and 7 green marbles. Then, the probability that 3 will be blue and 2 green is:a)213/429b)140/429c)117/429d)167/429Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Blue = 6
Green = 7
Number of ways 3 blue marbles can be drawn = ⁶C₃
number of ways 2 green marbles can be drawn = ₇C₂
total ways of drawing 5 marbles = ¹³C₅
=> probability = (⁶C₃ * ⁷C₂)/¹³C₅
= 140/429

Answer

Number of ways drawing blue marbles = 6C3
number of ways drawing green marbles = 7C2
total ways of drawing 5 marbles = 13C5

=> probability = 6C3*7C2/13C5 = 140/429

Answer

To solve this problem, we can use the concept of combinations and the probability of independent events.

Let's break down the problem step by step:

Step 1: Finding the total number of ways to draw 5 marbles from a bag of 6 blue marbles and 7 green marbles.
We can use the combination formula:
nCr = n! / (r!(n-r)!)
Here, n is the total number of marbles in the bag (13) and r is the number of marbles we want to draw (5).
So, the total number of ways to draw 5 marbles from the given bag is:
13C5 = 13! / (5!(13-5)!) = 13! / (5!8!) = (13*12*11*10*9) / (5*4*3*2*1) = 1287

Step 2: Finding the number of ways to draw 3 blue and 2 green marbles.
We need to select 3 blue marbles from the 6 available blue marbles, and 2 green marbles from the 7 available green marbles.
Therefore, the number of ways to draw 3 blue and 2 green marbles is:
6C3 * 7C2 = (6! / (3!(6-3)!)) * (7! / (2!(7-2)!))
= (6! / (3!3!)) * (7! / (2!5!))
= (6*5*4 / (3*2*1)) * (7*6 / (2*1))
= 20 * 21
= 420

Step 3: Finding the probability.
The probability of an event is given by the formula:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
In this case, the number of favorable outcomes (drawing 3 blue and 2 green marbles) is 420, as calculated in the previous step.
The total number of possible outcomes is 1287, as calculated in step 1.
Therefore, the probability is:
Probability = 420 / 1287 = 140/429

Hence, the correct answer is option B) 140/429.

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