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Question:

A local restaurant has 16 vegetarian items and 9 non-vegetarian items in their menu. Some items contain gluten, while the rest are gluten-free. 

One evening, Rohit and his friends went to the restaurant. They planned to choose two  different vegetarian items and three different non-vegetarian items from the entire menu. Later, Bela and her friends also went to the same restaurant: they planned to choose two different vegetarian items and one non-vegetarian item only from the gluten-free options. The number of item combinations that Rohit and his friends could choose from, given their plan, was 12 times the number of item combinations that Bela and her friends could choose from, given their plan. 

How many menu items contain gluten?

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).