Question 11

# 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?

Solution

The number of item combinations Rohit and his friends could have ordered veg items were $$^{16}C_2$$
And the number of ways they could have ordered non-veg items would be $$^9C_3$$

The total number of item combinations that Rohit and his friends could choose from would then be $$^{16}C_2\times\ ^9C_3$$

Let's denote the number of veg gluten-free items as $$V\sim G$$ and the number of non-veg gluten-free items as $$NV\sim G$$

The total number of ways Bela and her friends could have chosen the items from the menu would then be
$$NV\sim G\times\ ^{N\sim G}C_2$$

We are given that Rohit and his friend's number of combinations were twelve times this value, giving us the relation

$$^{16}C_2\times\ ^9C_3=12\times\ NV\sim G\times\ ^{N\sim G}C_2$$
$$^{16}C_2\times\ ^9C_3=12\times\ NV\sim G\times\ \frac{\left(N\sim G\right)\left(N\sim G\ -1\right)}{2}$$
$$2^4\times\ 3\times\ 5\times\ 7=\ NV\sim G\times\ \left(N\sim G\right)\left(N\sim G\ -1\right)$$

At this point, we need two consecutive numbers on the left-hand side to account for the $$\left(N\sim G\right)\left(N\sim G\ -1\right)$$ on the right-hand side.

We know that $$NV\sim G$$ must be 9 or less; trying to put this value as values from 9 to 1 and get two consecutive digits from the remaining numbers,

The value of $$NV\sim G$$ can be 7, giving the value of $$\left(N\sim G\right)\left(N\sim G\ -1\right)$$ as 16 x 15
OR
The value of $$NV\sim G$$ can be 8, giving the value of $$\left(N\sim G\right)\left(N\sim G\ -1\right)$$ as 15 x 14

In either of these cases, the number of items on the menu not having gluten would be 7+16 = 8+15 = 23
This means that 2 of the items on the menu will not have gluten in them.

Therefore, Option B is the correct answer.

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