Did I answer this problem right?

There are 6 contestants in a singing competition. How many different ways can first, second and third place be awarded?
Here my answer: 6![(6-3)!*3]=4*5*6/2*3=20
Is this correct?

No, it isn't.

I think the answer is 108 ... but I'm not exactly sure.

I don't think so.

There are people who can get first
six people who can get second
six peoplewho can get third.

Let the six folks be a, b, c, d, e, f

abc
abd
abe
abf
acb
acd
ace
acf
adb
adc
ade
adf
aeb
aec
aed
aef
afb
afc
afd
afe
so if a gets first, there twenty ways to get second and third.

So by the same logic, b,c,d,e,f can be first, so
ways: 20*6=120

check my thinking.

Your thinking is absolutely right. I made the mistake of counting only 18 ways for 2nd and 3rd (18*6 = 108).

Sry.

an easier way to think about it is that you have 6 ways to choose first place and considering that the same person can not get second place you only have 5 to choose from for second place and of course 4 people to choose from for third place.

that makes it 6*5*4 possibilities=120 different combinations for 1st,2nd and 3rd

There are 6 ways to fill first, then 5 ways to fill second, and 4 ways to hand out third prize.

So there are 6*5*4 or 120 ways, as Jordan said.

Except Jordan meant to say
120 different permutations and not combinations.

Permutations imply positioning, while combinations do not.

To determine if your answer is correct, let's analyze the problem step by step:

1. The number of ways to award first place is 6 because there are 6 contestants.

2. After awarding the first place, there are 5 remaining contestants eligible for the second place.

3. The number of ways to award the second place is therefore 5.

4. After awarding the first and second places, there are 4 remaining contestants eligible for the third place.

5. The number of ways to award the third place is 4.

To find the total number of different ways to award first, second, and third place, you need to multiply the number of possibilities for each rank:

6 (number of ways to award first place) * 5 (number of ways to award second place) * 4 (number of ways to award third place) = 120.

Therefore, there are 120 different ways to award first, second, and third place in this singing competition.

In your calculation, it seems you attempted to use factorials, but you made two errors:

1. You calculated 6![(6-3)!*3], which incorrectly incorporates a product of factorials involving the 6 contestants. Instead, you should have only used 6 for the first place, as explained above.

2. You divided the product 4 * 5 * 6 by 2 * 3, which is incorrect. This factor of 2 * 3 would only be relevant if you were trying to determine the number of ways to arrange the top three contestants within the awarded positions, disregarding their rank (e.g., ABC, ACB, BAC, etc.).

Therefore, your answer of 20 is not correct. The correct answer is 120.