Problem Solving

Five cards are to be selected at random from 10 cards numbered 1 to 10. How many ways are possible that the average of five numbers selected will be greater than the median?

OA 111

How do you do this in under 2 mins?

if median=3, we have to select 1,2,3. then selecting any 2 numbers other than 4,5 will result in mean > median. thus 7C2 - 1 = 20
if median=8, mean max=8. thus 0 way.
if median=7, mean max=8: sum of other 4=40-7 = 33. sum max can be: 5+6+9+10=30 0 way.
if median=6, mean 7: sum of other 4 numbers = 29. max = 4+5+9+10 = 28. not possible.
if median=6, mean =8: sum of other 4 numbers = 34. not possible
if median=5, mean=8: sum of other 4 = 35. not possible.
if median=5, mean=7: sum of other 4 = 30. not possible.
if median=5, mean=6: sum of other 4 = 25. max = 26. possible.
if sum of lowest 2=7, we have to choose 8 and 10.
if sum of lowest 2=6, we have to choose 9 and 10. 2 ways

if median=4, mean=8,7,6 not possible. mean=5 possible.
mean=5 and median=4, sum of other 4=21.
sum of lowest2 = 5, larger: (6,10)(7,9)
sum of lowest2 = 4, larger: (7,10)(8,9)
sum of lowest2 = 3, larger: (8,10)
5 ways. thus total = 20+2+5 = 27..

Cans, i m not sure if i understand your solution. It would be helpful if you can please elaborate on your explanation?

cans wrote:if median=3, we have to select 1,2,3. then selecting any 2 numbers other than 4,5 will result in mean > median. thus 7C2 - 1 = 20
if median=8, mean max=8. thus 0 way.
if median=7, mean max=8: sum of other 4=40-7 = 33. sum max can be: 5+6+9+10=30 0 way.
if median=6, mean 7: sum of other 4 numbers = 29. max = 4+5+9+10 = 28. not possible.
if median=6, mean =8: sum of other 4 numbers = 34. not possible
if median=5, mean=8: sum of other 4 = 35. not possible.
if median=5, mean=7: sum of other 4 = 30. not possible.
if median=5, mean=6: sum of other 4 = 25. max = 26. possible.
if sum of lowest 2=7, we have to choose 8 and 10.
if sum of lowest 2=6, we have to choose 9 and 10. 2 ways

if median=4, mean=8,7,6 not possible. mean=5 possible.
mean=5 and median=4, sum of other 4=21.
sum of lowest2 = 5, larger: (6,10)(7,9)
sum of lowest2 = 4, larger: (7,10)(8,9)
sum of lowest2 = 3, larger: (8,10)
5 ways. thus total = 20+2+5 = 27..

I posted a solution here:

https://www.beatthegmat.com/stats-probab ... 68548.html