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?

Accepted Answer

B. Total combinations for Rohit $= \binom{16}{2} 	imes \binom{9}{3} = 120 	imes 84 = 10080$. Let $V_{gf}$ be the number of gluten-free vegetarian items and $NV_{gf}$ be the number of gluten-free non-vegetarian items. Combinations for Bela $= \binom{V_{gf}}{2} 	imes \binom{NV_{gf}}{1} = \frac{V_{gf}(V_{gf}-1)}{2} 	imes NV_{gf}$. We are given $10080 = 12 	imes \frac{V_{gf}(V_{gf}-1) NV_{gf}}{2} \implies V_{gf}(V_{gf}-1) NV_{gf} = 1680$. Since $V_{gf} \le 16$ and $NV_{gf} \le 9$, let's test possible integer values of $NV_{gf}$ (from 9 downwards): If $NV_{gf} = 8$: $V_{gf}(V_{gf}-1) = 1680 / 8 = 210 = 15 	imes 14 \implies V_{gf} = 15$. Then total gluten-free items $= 15 + 8 = 23$. Since total menu items $= 16 + 9 = 25$, the number of items containing gluten $= 25 - 23 = 2$.

Menu Combinations and Gluten Items

Explanation

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