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?

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Answer 1

The Correct Option is B

Step 1: Calculate combinations for Rohit's group. Rohit's group selected 2 vegetarian items from 16 and 3 non-vegetarian items from 9:

$\binom{16}{2} \times \binom{9}{3} = \frac{16 \times 15}{2} \times \frac{9 \times 8 \times 7}{6} = 120 \times 84 = 10,080.$

Step 2: Calculate combinations for Bela's group. Let g represent the count of items containing gluten. The gluten-free items are calculated as:

(16 + 9) − g = 25 − g.

Bela's group chose 2 vegetarian and 1 non-vegetarian item from these gluten-free options:

$\binom{16 - g}{2} \times \binom{9 - g}{1}$

Step 3: Establish the relationship between the two combination counts. The problem states:

10,080 = 12 × $\left[ \binom{16 - g}{2} \times \binom{9 - g}{1} \right]$.

Simplifying this equation yields:

$\binom{16 - g}{2} \times \binom{9 - g}{1} = \frac{10,080}{12} = 840.$

Step 4: Determine the value of g. Expanding the combination formulas:

$\frac{(16 - g)(15 - g)}{2} \times (9 - g) = 840.$

Further simplification leads to:

$\frac{(16 - g)(15 - g)(9 - g)}{2} = 840 \implies (16 - g)(15 - g)(9 - g) = 1,680.$

By testing values for g, it is found that g = 3 satisfies the equation.

Answer: 3