Natural High School !!

Question

The students at Natural High School sell coupon books to raise money after-school programs. At the end of the coupon sale, the school selects six students to win prizes as follows: From the homeroom with the highest total coupon-book sales, the students with the first-, second- and third-highest sales receive $50, $30, $20, respectively; from the homeroom with the second-highest total coupon-book sales, the three highest-selling students receive $10 each. If Natural High School has ten different homerooms with eight student each, in how many different ways could the six prizes be awarded?

Do not calculate the actual number...Just the logic i wonder about...
Thnx

vineeshp · Answer

Challenging question, but it would be great if you post the full OA so we can work out a solution and verify our answer.

My understanding and Solution:
We can choose any one of the ten schools in 10 ways (10C1) and once a school is selected, there are 8 students to win the 1st three prices. So that can be done in 8P3 ways.
Hence 10 * 8P3 (=3360)

Next 2nd school can be selected from remaining 9 schools in 9 ways and each school has 3 identical winners (all get 10) which can be chosen in 8C3 ways. (504)

Now total number of ways of selecting the 6 winners is the product of the above 2.

Vineesh,
Just telling you what I know and think. I am not the expert.

Ozlemg · Answer

vineeshp wrote:Challenging question, but it would be great if you post the full OA so we can work out a solution and verify our answer.

My understanding and Solution:
We can choose any one of the ten schools in 10 ways (10C1) and once a school is selected, there are 8 students to win the 1st three prices. So that can be done in 8P3 ways.
Hence 10 * 8P3 (=3360)

Next 2nd school can be selected from remaining 9 schools in 9 ways and each school has 3 identical winners (all get 10) which can be chosen in 8C3 ways. (504)

Now total number of ways of selecting the 6 winners is the product of the above 2.

I do not have the exact number either. But Here is the steps.may they help to contsruct the formulas:

Break down the problem into 3 decisions you have to make:
1) Select 2 homerooms,1 with Highest sales and 1 with 2nd highest sales.(order matters here)
2)Select 3 Students from the highest sales Homeroom, again here order matters as depending on the position the value of the prize differs
3)select 3 students from the 2nd highest sales homeroom, here the order doesn’t matter as all 3 will receive the same prize.

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