Mean and Median

Expert replies

nakulbatra: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Sat Jan 31, 2009 10:16 am

by nakulbatra » Wed Feb 11, 2009 5:34 am

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 after some posts.

Like this post

Source: Beat The GMAT — Problem Solving | Wed Feb 11, 2009 5:34 am

hardik.jadeja: Legendary Member; Posts: 535; Joined: Fri Jun 08, 2007 2:12 amFollowed by: 5 members; Thanked: 87 times; GMAT Score:730

by hardik.jadeja » Thu Feb 12, 2009 2:24 pm

I am getting 49. Is 49 the answer? I am not very confident. But I can explain how i got 49 if its the correct answer.

Like this post

cjb: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Fri Oct 31, 2008 2:41 pm; Location: London, UK; Thanked: 5 times; GMAT Score:770

Re: Mean and Median

by cjb » Thu Feb 12, 2009 4:57 pm

nakulbatra wrote: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 after some posts.

My first attempt at this is:

10C5 = 252

For some of those picks of 5 cards, the mean will equal the median. The rest will be equally split between having a higher mean or a higher median. So we can solve if we know all the way in which mean and median will be the same.

Possible card distributions with equal mean and median is the problem. Can't think of a quick way of enumerating them.

For cards a,b,c,d,e, such that 1 <= a < b < c < d < e <=10
mean = (a+b+c+d+e)/5
median = c

(a+b+c+d+e)/5 = c
a+b+c+d+e = 5c

=> a+b+d+e =4c

(c-a)+(c-b) = (d-c)+(e-c)

Consider c=5. Max LHS of eqn above =
(5-1)+(5-2) = 7
This can be balanced on the RHS either with 7 and 10, or with 8 and 9.

I could probably get somewhere with this if I had infinite time, but I'd be way over the two minute mark now. Must be a trick I'm not seeing.

80% of success is showing up -- Woody Allen

Like this post

logitech: Legendary Member; Posts: 2134; Joined: Mon Oct 20, 2008 11:26 pmFollowed by: 25 members; Thanked: 237 times; GMAT Score:730

by logitech » Thu Feb 12, 2009 7:04 pm

For some of those picks of 5 cards, the mean will equal the median. The rest will be equally split between having a higher mean or a higher median. So we can solve if we know all the way in which mean and median will be the same.

Great start.

Median will be between 3 and 8 inclusive.

for 3 and 8 we only have one way to choose the rest of the 4 numbers

1 2 3 4 5

6 7 8 9 10

Between 3 and 8

4, 5, 6, and 7

For example FOR Median = Average = 4

2 3 4 5 6

We will have only (n-1) number on the left to play with 2 numbers and every time you make a change you have to make another change on the other side and you can choose 2 of them

For 4 = REST of the sum will be 16 ( you can use 1,2 or 3) C(3:2) = 3
For 5= REST of the sum will be 20 (you can use 1, 2, 3 or 4) C(4:2) = 6
For 6= REST of the sum will be 24 ( you can use 7,8,9 or 10 ) C(4:2) = 6
For 7= REST of the sum will be 28 ( you can use 8,9 or 10 ) C(3:2)=3

So AV=MEDIAN= 1+3+6+6+3+1 = 20 ways ( If I am not mistaken)

C(10:5)= 252

252 - 20 = 232 way for > Median and < Median

So answer should be 232/2 = 116

:roll:

I realized that I have missed couple of numbers!

median=3: only one way: 12345
median=4: four ways: 12467, 12458, 13457, 23456
median=5: ten ways: 12589, 1257T, 13579, 1356T, 14569, 14578, 23578, 23569, 24568, 34567
(I listed the numbers lower than the median systematically, then looked for pairs of numbers higher than the median that added to the right sum)
median=6 ten ways (symmetric with median=5)
median=7,four ways (symmetric with median=4),
median=8, 1 way 6789T
Total ways = 1+4+10+10+4+1 = 30

This makes the correct answer: 111

252-30 =222

and 222/2 = 111

LGTCH
---------------------
"DON'T LET ANYONE STEAL YOUR DREAM!"

Like this post

x2suresh: Master | Next Rank: 500 Posts; Posts: 258; Joined: Thu Aug 07, 2008 5:32 am; Thanked: 16 times

Re: Mean and Median

by x2suresh » Thu Feb 12, 2009 7:27 pm

nakulbatra wrote: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 after some posts.

Is this GMAT question..

If yes.. can you post 5 options too...

Like this post

nakulbatra: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Sat Jan 31, 2009 10:16 am

OA

by nakulbatra » Sun Feb 15, 2009 8:20 am

the OA is 111

Like this post

cjb: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Fri Oct 31, 2008 2:41 pm; Location: London, UK; Thanked: 5 times; GMAT Score:770

by cjb » Sun Feb 15, 2009 9:49 am

Now - what's the quick way of doing this?

80% of success is showing up -- Woody Allen

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review