In a basket ,there are 5 green , 4 blue and 2 red balls in a basket What is the probability of both balls being blue or green ball , if 2 balls are picked from the basket ?

Options

A. `(16)/(55)`
B. `(21)/(55)`
C. `(39)/(55)`
D. `(34)/(55)`

Answer

To solve the problem of finding the probability of picking two balls from a basket containing 5 green, 4 blue, and 2 red balls such that both balls are either blue or green, we can follow these steps:

### Step 1: Determine the total number of balls
First, we need to find the total number of balls in the basket.

- Green balls = 5
- Blue balls = 4
- Red balls = 2

Total balls = 5 + 4 + 2 = 11

### Step 2: Calculate the total number of ways to pick 2 balls
Next, we calculate the total number of ways to choose 2 balls from the 11 balls.

Using the combination formula \( nCk = \frac{n!}{k!(n-k)!} \):

\[ \text{Total ways to pick 2 balls} = 11C2 = \frac{11!}{2!(11-2)!} = \frac{11 \times 10}{2 \times 1} = 55 \]

### Step 3: Calculate the number of favorable outcomes for blue balls
Now, we calculate the number of ways to pick 2 blue balls from the 4 blue balls.

\[ \text{Ways to pick 2 blue balls} = 4C2 = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Calculate the number of favorable outcomes for green balls
Next, we calculate the number of ways to pick 2 green balls from the 5 green balls.

\[ \text{Ways to pick 2 green balls} = 5C2 = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the total number of favorable outcomes
Now, we add the number of ways to pick 2 blue balls and 2 green balls to find the total favorable outcomes.

\[ \text{Total favorable outcomes} = \text{Ways to pick 2 blue balls} + \text{Ways to pick 2 green balls} = 6 + 10 = 16 \]

### Step 6: Calculate the probability
Finally, we can calculate the probability of picking 2 balls that are either blue or green.

\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{16}{55} \]

Thus, the probability of both balls being blue or green is \( \frac{16}{55} \).

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