Discussion

Explanation:

Let the number of gluten free - vegetarian items be x and number of gluten free - non-vegetarian items be y.

∴ We have the following

Vegetarian items (16)

• Gluten free = x
• Gluten free = 16 - x

Non-Vegetarian items (9)

• Gluten free = y
• Gluten free = 9 - x

Rohit & his friends planned to choose two different vegetarian items and three different non-vegetarian
Number of ways of doing this = ¹⁶C₂ × ⁹C₃ = 120 × 84

Bela & his friends planned to choose two different vegetarian items and one non-vegetarian only from gluten-free options
Number of ways of doing this = ^xC₂ × ^yC₁ = x(x - 1)/2 × y

According to the question:
120 × 84 = 12 × x(x - 1)/2 × y
⇒ 120 × 84 = 6 × x(x - 1) × y
⇒ 20 × 84 = x(x - 1) × y
​​​​​​​⇒ 20 × 84 = x(x - 1) × y
⇒ 1680 = x(x - 1) × y

Here, x ≤ 16 & y ≤ 9

​​​​​​​Case 1: x = 16
⇒ 1680 = 16(16 - 1) × y
⇒ y = 7

Case 2: x = 15
⇒ 1680 = 15(15 - 1) × y
⇒ y = 8

Case 3: x = 14
⇒ 1680 = 14(13 - 1) × y
⇒ y = 9.23 (rejected)

We will note consider further cases for x, since y will be greater than 9, which is not possible.

∴ We have two cases for (x, y) = (16, 7) & (15, 8)

In both these cases, number of gluten free items = 16 + 7 or 15 + 8 = 23

∴ Out of total 16 + 9 = 25 items, 23 are definitely gluten free, hence 2 items will contain gluten.

Hence, option (b).

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