Question

When making a fruit salad, a chef chooses 7 fruits from 10 that are in season. how many different fruit salads can the chef create?

Answer 1

For this example, we must combine not permutate; this because a salad in which you first use apple then tomato is the same as the one in which you use first tomato then apple, so order does not matter.

Me must do "10 combine 7" which is written by the binomial coefficient $(_{7} ^{10})$; in general a binomial coefficient is written by:
$(_{k} ^{n})= \frac{n!}{k!(n-k)!}$

In which $n$ and $k$ are positive integers, such that $n\ \textgreater \ k$.

So, for this problem:
$(_{7} ^{10})= \frac{10!}{7!(10-7)!}=\frac{10!}{7!3!}= \frac{10*9*8}{3*2}=10*3*4=120$

Hence, there are 120 different ways to make the fruit salad.