Explanation:

Let’s break this problem into bite-sized pieces:

- Total blue balls in all boxes = 9

- Total balls across P, Q, R = 32

- Box P: Probability(Blue) = 1/4 ? Blue balls in P = Bp, so Bp/Np = 1/4 ? Np = 4Bp

- Red balls in P = Bp + 2

- Box Q: Total balls = 10. Let Blue in Q = Bq, Red in Q = Rq, Green in Q = Gq

- Probability ratio Blue:Red = 2:3 ? Bq/Rq = 2/3

- Bq + Gq = Rq + 4

- Box R: Total balls = 10. Probability(Red) is 1/5 less than Probability(Blue)

- Let Blue in R = Br, Red in R = Rr, Green in R = Gr

- Rr/10 = Br/10 – 1/5 ? Rr = Br – 2

Now, step-by-step math:

Box P:

- Let Blue balls in P = x

- Np = 4x

- Red balls = x + 2

- So Green balls = Np – (x + x + 2) = 4x – (x + x + 2) = 2x – 2

Box Q:

- Bq + Rq + Gq = 10

- Bq/Rq = 2/3 ? 3Bq = 2Rq ? Bq = (2/3)Rq

- Let Rq = y, then Bq = (2/3)y

- Bq + Gq = Rq + 4 ? (2/3)y + Gq = y + 4 ? Gq = y + 4 – (2/3)y = (1/3)y + 4

So, sum:

- (2/3)y + y + (1/3)y + 4 = 10

- (2/3 + 1 + 1/3)y + 4 = 10

- (2y) + 4 = 10 ? 2y = 6 ? y = 3

Thus, Rq = 3, Bq = 2, Gq = 5

Box R:

- Br + Rr + Gr = 10

- Rr = Br – 2

- So (Br – 2) + Br + Gr = 10 ? 2Br – 2 + Gr = 10 ? Gr = 12 – 2Br

We need to check all possibilities for blue balls in each box.

Total Blue balls = x (Box P) + 2 (Box Q) + Br (Box R) = 9

Let’s try:

x + 2 + Br = 9 ? x + Br = 7

From Box P, x must be integer and Np = 4x = 32 (Since total balls = 32), so try x values.

Try x = 3 ? Br = 4

Box P:

x = 3

Np = 12

Red = 5

Green = 12 – 3 – 5 = 4

Box Q:

Blue = 2, Red = 3, Green = 5

Box R:

Br = 4, Rr = 4 – 2 = 2, Gr = 10 – 4 – 2 = 4

Check total balls:

Box P: 12

Box Q: 10

Box R: 10

Sum = 12 + 10 + 10 = 32

Blue balls: 3 (P) + 2 (Q) + 4 (R) = 9

Everything fits. Now, find probability that all three are red:

- P (red in P) = Red balls in P / Total balls in P = 5/12

- P (red in Q) = 3/10

- P (red in R) = 2/10 = 1/5

So, overall probability = 5/12 × 3/10 × 1/5 = (5×3×1)/(12×10×5) = 15/600 = 1/40

Here’s what this means:

- Every part of the question lines up—each step flows into the next.

- The options are about calculating this "all red" scenario from the setup.

- Option 1 (1/40) is exactly the correct answer.

Correct Answer: Option 1 — 1/40