Hello sandy

In questions such as between so and so value such as 5 and 8 is this always the case that the two values are inclusive?

In not is specified you should consider 5 and 8 included. I.E the range is not from 6 to 7 but 5 to 8.

This is a good question for practice but not the top notch. An official question must be included in the statement if they should be considered or not. To clear all the possible confusion.

Regards

Whenever it is written from a to b, both ends must be included, but when it is written between a and b, ends are not included, its b-a-1 no of terms. Therefore, in this case its just 6 and 7.
If anyone says this then he or she is wrong. If not mentioned whether it is inclusive or not, it must be assumed inclusive.

Given that the contestants at a baking contest must use between 5 and 8 of 10 possible ingredients.

Now the number of ingredients which we can choose can be between 5 and 8(assuming inclusive) => 5, 6, 7, 8

If we are choosing 5, then we can choose 5 out of 10 on 10C5 ways = \(\frac{10!}{(5!)*(10-5)!}\) = \(\frac{10!}{5!*5!}\) = \(\frac{10*9*8*7*6*5!}{5!*1*2*3*4*5}\) = 252 ways

If we are choosing 6, then we can choose 6 out of 10 on 10C6 ways = \(\frac{10!}{(5!)*(10-6)!}\) = \(\frac{10!}{6!*4!}\) = \(\frac{10*9*8*7*6!}{6!*1*2*3*4}\) = 210 ways

If we are choosing 7, then we can choose 7 out of 10 on 10C7 ways = \(\frac{10!}{(7!)*(10-7)!}\) = \(\frac{10!}{7!*3!}\) = \(\frac{10*9*8*7!}{7!*1*2*3}\) = 120 ways

If we are choosing 8, then we can choose 8 out of 10 on 10C8 ways = \(\frac{10!}{(8!)*(10-8)!}\) = \(\frac{10!}{8!*2!}\) = \(\frac{10*9*8!}{8!*1*2}\) = 45 ways

Quantity A: The number of ingredients that must be used to get the smallest possible number of different combinations
Smallest number is produced when we take number of ingredients as 8
=> Quantity A = 8

Quantity B: The number of ingredients that must be used to get the largest possible number of different combinations.
Greatest number is produced when we take number of ingredients as 5
=> Quantity B = 5

Clearly, Quantity A > Quantity B

So, Answer will be A
Hope it helps!