Combination

[Abdulla]

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? (Assume that there are no ties, either among students or among homerooms.) Write your answer as a product of primes raised too various power ( do not actually compute the number).

OA is [spoiler](2^8)(3^3)(5)(7^2)[/spoiler]

[uttam.albela]

Hi Abdulla,

There are 10 homerooms. One of them will be with highest total.

Number of ways to select one from ten homerooms. = 10C1 = 10

Now from its 8 students select 3 = 8C3
As 3 awards are of different denominations(50,30 and 20), awards can be distributed among the 3 students in = 3!

Now select 2nd homeroom from 9 left = 9C1
Select 3 student from this selected homeroom = 8C3
As 3 awards are of the same denomination of 10 each, number of ways to distribute 3 awards among 3 students = 1

Multiply all these number to get the answer.

Do ask me if anything is not clear to you.

[sanjana]

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 doesnt matter as all 3 will receive the same prize.

Decision 1) No of ways : 10p2 = 10!/8! = 10*9 = 2*5*3*3

Decision 2) No of ways : 8p3 = 8!/5! = 8*7*6 =
2*2*2*7*3*2

Decision 3) No of ways : 8C3 = 8!/3!*5! = 8*7 = 2*2*2*7

Multiply all : we get the answer.