[Velvet]

A music festival award gives awards for 1st, 2nd and 3rd place in each category. If there are 7 contestants for piano and 9 for violin, in how many different ways could the 6 awards be presented?

[retiredguy123]

I don't understand the question. Wouldn't you give the awards to the 3 best contestants in each category? So, I guess my answer would be that there is only one way that is fair and makes sense.

[Velvet]

There is 1st, 2nd and 3rd place for piano, and 1st, 2nd and 3rd place for violin. Well, the question also assumes that even the worst player could get 1st.

[tuccillo]

I believe what you are looking for are the number of permutations. If that is the case, the number is 210 for the first category and 504 for the second category.

Originally Posted by Velvet

A music festival award gives awards for 1st, 2nd and 3rd place in each category. If there are 7 contestants for piano and 9 for violin, in how many different ways could the 6 awards be presented?

[retiredguy123]

LOL. So, the awards are randomly selected? I could solve it, but I have had too many adult beverages.

I think the best player should get first place. Why would you give the worst player an award? Not a math problem. Sounds like woke problem.

[Velvet]

I took the problem from a school math test. The question is looking for the total possibilities…( and that’s where I made a mistake when I tried it. )

[tuccillo]

If you want the number of permutations of 6 winners (when considering only 1-2-3 place within each of the two categories), I believe the number is 105,840.

Originally Posted by Velvet

I took the problem from a school math test. The question is looking for the total possibilities…( and that’s where I made a mistake when I tried it. )

[Velvet]

Originally Posted by tuccillo

If you want the number of permutations of 6 winners (when considering only 1-2-3 place within a group), I believe the number is 105,840.

Yes! That’s where I made my mistake, I added instead of multiplied.(Standardized grade 7 math test.)

[Davonu]

(7x6x5)x(9x8x7)

What Tuccillo said.

[tuccillo]

Interesting approach, just divide out the denominator.

For those who are curious, the general formula for permutations is:

n! / (n-s)!

where n is the number of objects you are considering and s is the number of objects in each set. Apply this to each of the two categories and then multiply.

Originally Posted by Davonu

(7x6x5)x(9x8x7)

What Tuccillo said.

[Keefelane66]

First prize can go to any 1 of the 15 participants, So 15 choices.

2nd prize can then go to 1 out of the remaining 14.

And 3rd can go to any one of the remaining 13.

So number of ways we can do this is 15*14*13

3*5*2*7*13

3*10*91

2730 ways.

[redwing1]

Originally Posted by Velvet

A music festival award gives awards for 1st, 2nd and 3rd place in each category. If there are 7 contestants for piano and 9 for violin, in how many different ways could the 6 awards be presented?

For the piano (7 total participants): There are 210 possible arrangements (permutations) of 3 participants. 7 possibilities for 1st Place, 6 possibilities for 2nd place, and 5 possibilities for 3rd place. Multiplying these (7x6x5) gives you 210 possible arrangements.

For the Violin (9 total participants) : There are 504 possible arrangements (permutations) of 3 participants. 9 possibilities for 1st Place, 8 possibilities for 2nd place, and 7 possibilities for 3rd place. Multiplying these (9x8x7) gives you 504 possible arrangements.

So, the total number of ways the trophies can be presented is 7 x 6 x 5 x 9 x 8 x 7 = 105,840.

[ In other words the piano 1st place trophy would be given to one of seven participants; the piano 2nd place trophy would be given to one of the remaining six participants, and the piano 3rd place trophy would be given to one of the remaining 5 participants. Similarly, the violin 1st place trophy would be given to one of nine participants; the violin 2nd place trophy would be given to one of the remaining eight participants; and the violin 3rd place trophy would be given to one of the remaining 7 participants. ]

[Two Bills]

My grandson loves math.
I can't even understand any of the stuff he does.
Kids today are so far in front of where we were at same age.
I was a bit thick anyway, but I knew how to work out my hours and wages!

[Normal]

It’s a perm, because order does matter. Don’t treat it as a combination.

[Arctic Fox]

Originally Posted by tuccillo

If you want the number of permutations of 6 winners (when considering only 1-2-3 place within each of the two categories), I believe the number is 105,840.

another vote for Tuccillo's answer

As an aside, my maternal grandmother's village in England held an annual Show and one year refused to give a 1st Place for the carrots as none of the entries was considered good enough. They just awarded 2nd and 3rd.