Question

Problem 10: In a group of 10 people, 3 prizes will be awarded. How many ways can the prizes be distributed?

Answer 1

From the above group of 10 people, 3 prizes can be awarded in 120 different ways.

For calculating the number of ways the prizes can be distributed among the 10 people, we should use the concept of combinations because in this case, we need to choose 3 people out of 10 to receive the prizes. The order in which the prizes are awarded does not matter, so we should use combinations instead of permutations.

The number of ways to choose 3 people out of 10 can be calculated using the binomial coefficient formula:

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

Applying this formula, we have:

`C(10,3)=10!/(3!(10-3)!)\\`

`=10!/(3!` ×
`7!)`

`=10` ×
`9` ×
`8` ×
`7`! / (
`3` ×
`2` ×
`1` ×
`7`! )

`=10` ×
`9` ×
`8` / (
`3` ×
`2` ×
`1` )

`=120`

Therefore, there are 120 different ways in which 3 prizes can be distributed among a group of people of 10

Answer 2

if each person can get more than 1 prize,

it's 10^3 = 1000

if not

10 choose 3 = 10 * 9 * 8 / (3*2) = 120