8 trophies can be awarded to 30 people in how many different ways

Assuming the trophies are identical, or must be awarded in order 1st-to-last, then we just need to count the ways of choosing 8 people from 30.

C(30,8) = 5,852,925

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To find the number of different ways to award 8 trophies to 30 people, we can use the concept of combinations. Specifically, we can use the combination formula, which is given by:

C(n, r) = n! / (r! * (n-r)!)

Where n is the total number of items (trophies in this case) and r is the number of items chosen (people receiving the trophies). The exclamation mark denotes factorials.

In this scenario, we need to find the number of ways to choose 8 people out of a group of 30 to receive the trophies. Therefore, we can substitute n = 30 and r = 8 into the combination formula:

C(30, 8) = 30! / (8! * (30-8)!)

To calculate this, we simply need to evaluate the factorials:

30! = 30 * 29 * 28 * ... * 3 * 2 * 1
8! = 8 * 7 * 6 * ... * 3 * 2 * 1
(30-8)! = 22!

Simplifying further:

C(30, 8) = 30! / (8! * 22!)

Calculating the factorial values:

C(30, 8) = 30 * 29 * ... * 23 * 22! / (8 * 7 * ... * 2 * 1 * 22!)

Notice that the term 22! appears on both the numerator and denominator, and they cancel out:

C(30, 8) = 30 * 29 * ... * 23 / (8 * 7 * ... * 2 * 1)

Now, we can calculate this expression:

C(30, 8) = 3,435,036

Therefore, there are 3,435,036 different ways to award 8 trophies to 30 people.