A bag has 15 red, green, and blue balls. The number of each color ball is different. The difference between the number of red balls and green balls is the same as the difference between the number of green balls and blue balls. The probability of selecting one blue ball from the bag is greater than 0.2. Which of the following can be the number of blue balls in the bag?

Options:

(A) 3

(B) 4

(C) 5

(D) 7

(E) 9

Question

A bag has 15 red, green, and blue balls. The number of each color ball is different. The difference between the number of red balls and green balls is the same as the difference between the number of green balls and blue balls. The probability of selecting one blue ball from the bag is greater than 0.2. Which of the following can be the number of blue balls in the bag?

Options:

(A) 3

(B) 4

(C) 5

(D) 7

(E) 9

Accepted Answer

**Solution:**

1. Let the number of red, green, and blue balls be $x$, $y$, and $z$ respectively.

2. Given the difference between red and green balls is the same as the difference between green and blue balls:
   
   $$x - y = y - z$$

Rearranging:

$$2y = x + z$$

3. Total balls are 15:

$$x + y + z = 15$$

4. From step 2, express $x$ in terms of $y$ and $z$:

$$x = 2y - z$$

5. Substitute into total balls equation:

$$2y - z + y + z = 15 \ 3y = 15 \ y = 5$$

6. So, green balls $y = 5$.

7. Since $x = 2y - z = 10 - z$, and all numbers are different, $x$, $y$, and $z$ are distinct positive integers.

8. Probability of selecting a blue ball is greater than 0.2:

$$\frac{z}{15} > 0.2 \ z > 3$$

9. Also, $x$, $y$, $z$ are distinct positive integers, so check possible $z$ values greater than 3:

- If $z=4$, then $x=10 - 4 = 6$ (distinct from 5 and 4)
   - If $z=5$, then $x=10 - 5 = 5$ (not distinct)
   - If $z=7$, then $x=10 - 7 = 3$ (distinct)
   - If $z=9$, then $x=10 - 9 = 1$ (distinct)

10. Check which $z$ values are possible from the options:

- 3: Not possible since $z > 3$
    - 4: Possible
    - 5: Not possible (x and y equal)
    - 7: Possible
    - 9: Possible

11. Therefore, the possible number of blue balls are 4, 7, and 9.

12. From the options given, only (B), (D), and (E) correspond to these values.

**Answer:** Only (B), (D), and (E)

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